Transcript
Page 1: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Laplace Transforms for Systems ofDifferential Equations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 2: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System

1. When you have several unknown functions x,y, etc., thenthere will be several unknown Laplace transforms.

2. Transform each equation separately.3. Solve the transformed system of algebraic equations for

X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any

order.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 3: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then

there will be several unknown Laplace transforms.

2. Transform each equation separately.3. Solve the transformed system of algebraic equations for

X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any

order.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 4: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then

there will be several unknown Laplace transforms.2. Transform each equation separately.

3. Solve the transformed system of algebraic equations forX,Y, etc.

4. Transform back.5. The example will be first order, but the idea works for any

order.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 5: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then

there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for

X,Y, etc.

4. Transform back.5. The example will be first order, but the idea works for any

order.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 6: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then

there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for

X,Y, etc.4. Transform back.

5. The example will be first order, but the idea works for anyorder.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 7: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

The Laplace Transform of a System1. When you have several unknown functions x,y, etc., then

there will be several unknown Laplace transforms.2. Transform each equation separately.3. Solve the transformed system of algebraic equations for

X,Y, etc.4. Transform back.5. The example will be first order, but the idea works for any

order.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 8: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 9: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 10: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t)

OriginalDE & IVP

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 11: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t)

OriginalDE & IVP

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 12: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 13: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 14: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 15: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solution

-L

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 16: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solution

-

L

L −1

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 17: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

So Everything Remains As It Was

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solutionSolution

-

L

L −1

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 18: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.5. The example itself is related to equations that come from

the analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 19: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.5. The example itself is related to equations that come from

the analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 20: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.5. The example itself is related to equations that come from

the analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 21: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.5. The example itself is related to equations that come from

the analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 22: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.

5. The example itself is related to equations that come fromthe analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 23: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

1. Note that the second equation is not really a differentialequation.

2. This is not a problem. The transforms will work the sameway, ...

3. ... but the second equation also relates the initial values toeach other. So certain combinations of initial values willnot be possible.

4. The initial values in this example are allowed.5. The example itself is related to equations that come from

the analysis of two loop circuits. So systems such as thisone certainly arise in applications.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 24: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 25: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 26: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+1

2X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 27: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0

(×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 28: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 29: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y

(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 30: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s

+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 31: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4)

=2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 32: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12

=12s+14

s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 33: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 34: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

6x+6y′+ y = 2e−t, x(0) = 12x− y = 0, y(0) = 2

6X +6sY−12+Y =2

s+12X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 35: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 36: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 37: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)

s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 38: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 :

2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 39: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2

= A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 40: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2)

, A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 41: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 42: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

:

6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 43: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6

= B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 44: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

)

, B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 45: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 46: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4

=− 1s+1

+31

s+ 23

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 47: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 48: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

12s+14(s+1)(6s+4)

=A

s+1+

B6s+4

12s+14 = A(6s+4)+B(s+1)s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

13

), B = 18

Y(s) = − 1s+1

+18

6s+4=− 1

s+1+3

1s+ 2

3

y(t) = −e−t +3e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 49: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]= −1

21

s+1+

32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 50: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]= −1

21

s+1+

32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 51: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]= −1

21

s+1+

32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 52: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]

= −12

1s+1

+32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 53: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]= −1

21

s+1+

32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 54: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Solve the Initial Value Problem6x+6y′+ y = 2e−t, 2x− y = 0, x(0) = 1, y(0) = 2

2X−Y = 0

X =12

Y

=12

[− 1

s+1+3

1s+ 2

3

]= −1

21

s+1+

32

1s+ 2

3

x(t) = −12

e−t +32

e−23 t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 55: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 56: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values:

Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 57: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!

2x− y = 0: Same as x =y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 58: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0:

Same as x =y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 59: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

.

Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 60: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 61: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y =

6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 62: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 63: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 64: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 65: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t

(−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 66: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3

+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 67: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6

−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 68: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1)

+ e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 69: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t

(9−12+3)= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 70: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9

−12+3)= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 71: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12

+3)= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 72: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 73: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations

Page 74: Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check Laplace Transforms for Systems of Differential Equations Bernd Schroder¨ Bernd Schroder¨

logo1

New Idea An Example Double Check

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

23 t

Really Solve the Initial Value Problem6x+6y′+y = 2e−t, 2x−y = 0, x(0) = 1, y(0) = 2

Initial values: Look at x and y!2x− y = 0: Same as x =

y2

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

32

e−23 t]

+6[e−t−2e−

23 t]

+[−e−t +3e−

23 t]

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms for Systems of Differential Equations


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