Laplace Transforms for Systems of Differential Equationslogo1 New Idea An Example Double Check...

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

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.

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

order.

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.

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.

X,Y, etc.4. Transform back.

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

order.

So Everything Remains As It Was

Time Domain (t)

OriginalDE & IVP

Time Domain (t)

OriginalDE & IVP

Time Domain (t)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic solution,partial fractions

OriginalDE & IVP

Laplace transformof the solution

OriginalDE & IVP

Laplace transformof the solution

L −1

OriginalDE & IVP

Laplace transformof the solutionSolution

L −1

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.

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.

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)

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)

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

6X +6sY−12+Y =2

2X−Y = 0 (×−3 and add)

Y(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

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)

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)

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)

(6s+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

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)

+4) =2

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

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)

s+1+12 =

12s+14s+1

Y =12s+14

(s+1)(6s+4)

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

Y =12s+14

(s+1)(6s+4)

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)

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)

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−1 : 2 = A(−2), A =−1

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

12s+14 = A(6s+4)+B(s+1)s =−1 :

2 = A(−2), A =−1

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

12s+14 = A(6s+4)+B(s+1)s =−1 : 2

= A(−2), A =−1

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

, A =−1

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

, B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

=− 1s+1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

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

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

s =−23

: 6 = B(

), B = 18

Y(s) = − 1s+1

6s+4=− 1

y(t) = −e−t +3e−23 t

2X−Y = 0

[− 1

]= −1

x(t) = −12

e−t +32

e−23 t

2X−Y = 0

[− 1

]= −1

x(t) = −12

e−t +32

e−23 t

2X−Y = 0

[− 1

]= −1

x(t) = −12

e−t +32

e−23 t

2X−Y = 0

[− 1

= −12

x(t) = −12

e−t +32

e−23 t

2X−Y = 0

[− 1

]= −1

x(t) = −12

e−t +32

e−23 t

2X−Y = 0

[− 1

]= −1

x(t) = −12

e−t +32

e−23 t

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

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 =

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

Initial values:

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

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

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 +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

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 +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y =

6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t

(−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3

+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6

−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1)

+ e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t

(9−12+3)= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9

−12+3)= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12

+3)= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t

Does x(t) =−12

e−t +32

e−23 t, y(t) =−e−t +3e−

. Look at x and y!

6x+6y′+ y = 6[−1

2e−t +

e−23 t]

+6[e−t−2e−

+[−e−t +3e−

= e−t (−3+6−1) + e−23 t (9−12+3)

= 2e−t √

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