Upload
rootveshmehta
View
220
Download
0
Embed Size (px)
Citation preview
8/12/2019 Final Lecture 6 Maths-3
1/35
Higher Order Linear Differential
Equations
Prepared By-Rootvesh Mehta
1Sci.& Hum.Dept. ,E.M.-319/07/2013
8/12/2019 Final Lecture 6 Maths-3
2/35
Basic concepts
Definition-
Differential Equation -- A differential equation
is an equation containing an unknown
function and its derivatives. Examples are
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 2
36
4
3
3
y
dx
dy
dx
yd
0z z
x y
8/12/2019 Final Lecture 6 Maths-3
3/35
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 3
Differential equations can be
classified in two parts
(1) Ordinary differential equations
(2) Partial differential equations
An Ordinary differential equation is anequation which involves ordinary derivatives
is an example of O.D.E
A partial differential equation is an equationwhich involves partial derivatives
Is an example of P.D.E
36
4
3
3
y
dx
dy
dx
yd
0z z
x y
8/12/2019 Final Lecture 6 Maths-3
4/35
Order of Differential Equation
The order of the differential equation is the
order of the highest order derivative in the
differential equation
Differential Equation ORDER
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 4
32 xdx
dy
1
0932
2
ydx
dy
dx
yd 2
8/12/2019 Final Lecture 6 Maths-3
5/35
Degree of Differential Equation
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 5
The degree of a differential equation is power of the highest order
derivative term in the differential equation.
Differential Equation Degree
032
2
aydx
dy
dx
yd
03
53
2
2
dxdy
dxyd
1
3
8/12/2019 Final Lecture 6 Maths-3
6/35
Solution of a differential equation
Definition--The solution of a differential
equation is a function which satisfies given
equation
types of solutions of differential equations
1) General solution
2) Particular Solution
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 6
8/12/2019 Final Lecture 6 Maths-3
7/35
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 7
General Solution- The solution of differential equation
is called general if the no. of arbitrary constants equals
to the order of differential equationParticular solution- If we assign particular value to
arbitrary constant in general solution then it is called
particular solution
Example -y=3x+c is solution of the 1st
order differentialequation ,here its a general solution
Now ,if we take c=5 in y=3x+c then its a particularsolution.
8/12/2019 Final Lecture 6 Maths-3
8/35
Linear differential equations
A differential equation is called linear if
unknown function and the derivative of
unknown function are of first degree and they
are not multiplied together
A Differential equation of the form
is called linear differential equation of the first
order where either P and Q are functions of x
or constant
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 8
dyPy Q
dx
8/12/2019 Final Lecture 6 Maths-3
9/35
Solution of first order Linear
differential equations
The solution of first order linear diff.eqn. can
be obtained as follows ,
Integrating factor is
The General solution is
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 9
Pdx
e
( . .) ( . .)y I F Q I F dx c
8/12/2019 Final Lecture 6 Maths-3
10/35
General Form of Higher order Linear
Differential Equations
General form of second order Linear Differentialequation is
where P and Q are functions of x or constants
General form of nthorder Linear Differential equationis
--Where are constants or functions of x or
constants
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 10
2
2
d y dy P Qy R
dx dx
1 2
0 1 2 11 2 .........
n n n
n nn n n
d y d y d y dy a a a a y b
dx dx dx dx
0 1 1, ,.........., ,
na a a b
8/12/2019 Final Lecture 6 Maths-3
11/35
Classification of Higher order Linear
Differential Equations
Higher orderLinear diff.Eqns
HomogenousLinear diff.Eqns
ConstantCoefficients
Variablecoefficients
Non-homogenousLinear diff.Eqns
ConstantCoefficients
VariableCoefficients
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 11
8/12/2019 Final Lecture 6 Maths-3
12/35
General Form of Higher order Linear Differential
Equations
General form of second order Linear Differentialequation is
where P and Q are functions of x or constants
General form of nthorder Linear Differentialequation is
--
Where are constants or functions of xor constants
19/07/2013 12
2
2
d y dy P Qy R
dx dx
Sci.& Hum.Dept. ,E.M.-3
1 2
0 1 2 11 2 .........
n n n
n nn n n
d y d y d y dy a a a a y b
dx dx dx dx
0 1 1, ,.........., ,
na a a b
8/12/2019 Final Lecture 6 Maths-3
13/35
Note---
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 13
This is another form of nth order lineardifferential equation
-----------------(1)where are functions of x or constants.Here in eqn-1 left hand side we have n+1coefficients but we will divide eqn-1 by andtherefore we get there n-coefficients so in lineardiff.eqn the no. of coefficients is equal to theorder of diff.eqn
1
0 1 11 .........
n n
n nn n
d y d y dy a a a a y b
dx dx dx
0 1 1, ,.........., ,,
n na a a a b
0a
8/12/2019 Final Lecture 6 Maths-3
14/35
Homogenous and Non-homogenous
Linear differential Eqn. of higher Order
----------(1)
If b = 0in Eqn-1 then it is called Homogenouslinear differential eqn. And b is non-zero in
Eqn-1 that means either b is a function of x or
constant then it is called non-homogenous
linear differential eqn.
19/07/2013 14
1 2
0 1 11 2 .........
n n n
nn n n
d y d y d y dy a a a b
dx dx dx dx
Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
15/35
Homogenous and Non-homogenous Linear differential Eqn. of
higher Order with constant and variable Coefficients
if are functions of x in the following
equation
then that equation is calledLinear differential eqn.with variable coefficients.
If are constants in given eqn.then it is
called Linear differential eqn. of higher Order withconstant coefficients.
19/07/2013 15
1 2
0 1 11 2 .........
n n n
nn n n
d y d y d y dy a a a b
dx dx dx dx
0 1 1, ,.........., ,
na a a b
0 1 1, ,.........., ,
na a a b
Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
16/35
Linear combination of functions
Let are functions and
are constants then the expression
Is called linear combination of functions
19/07/2013 16Sci.& Hum.Dept. ,E.M.-3
1 1 2 2( ) ( ) ...... ( )
n nf x c f x c f x c
1 2, , ......., nc c c
1 2( ), ( ), .... ( )
nf x f x f x
1 2( ), ( ), .... ( )
nf x f x f x
8/12/2019 Final Lecture 6 Maths-3
17/35
Linear independent and Dependent
functions
19/07/2013 17
Let are functions and
are constants and If
and all
then the given functions are called linearlyindependent functions and if
and at least one then the givenfunctions are called linearly dependent functions
1 2( ), ( ), .... ( )
nf x f x f x
1 2, , ......., nc c c
1 1 2 2 0( ) ( ) ...... ( )
n nf x c f x c f x c
1 2 0
, , ......., nc c c
1 1 2 2 0( ) ( ) ...... ( )n nf x c f x c f x c 0
ic
Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
18/35
Wronskian Test for Linearly
Independent -Dependent
19/07/2013 18Sci.& Hum.Dept. ,E.M.-3
Let are n functions and
their first n-1 derivatives exists then the
Wronskian or Wronski Determinant is denoted
by and defined as
1 2( ), ( ), .... ( )
nf x f x f x
1 2, , ........,( )
nW f f f
1
1
1
1 1
1
' '
'' ''
( ) ( )
....
....
....
....................
....................
.......
n
n
n
n n
n
f f
f f
f f
f f
8/12/2019 Final Lecture 6 Maths-3
19/35
Wronskian Test
If then given functions
are Linearly Independent and
if are solutions of given differentialequation and
then given functions are linearly dependent but for functions whichare not solutions of given Diff.Eqn and
then they may or may not be linearly dependent
19/07/2013 Sci.& Hum.Dept. ,E.M.-3 19
1 2 0, , ........,( )nW f f f
1 2( ), ( ), .... ( )
nf x f x f x
1 2( ), ( ), .... ( )
nf x f x f x
1 2 0, ,........,( )nW f f f
1 2 0, ,........,( )nW f f f
8/12/2019 Final Lecture 6 Maths-3
20/35
Example
19/07/2013 20Sci.& Hum.Dept. ,E.M.-3
Using Wronskian test check whether given
functions f(x)=sinx and g(x) =cosx are linearly
independent or dependent?
Using Wronskian test check whether given
functions
are linearly independent or dependent?
( ) , ( )x x
f x e g x e
8/12/2019 Final Lecture 6 Maths-3
21/35
Solution of homogenous linear
differential equations of higher order
19/07/2013 21Sci.& Hum.Dept. ,E.M.-3
Superposition Principle for Linearity
8/12/2019 Final Lecture 6 Maths-3
22/35
Find Second Solution if one is
given(Method of reduction of order)
If is the one solution of second order
homogenous linear differential equation
Then second sol. where
this method is also known as method of
reduction of order
19/07/2013 22
1y
0y py qy
2 1y uy
2
1
1 pdxu e dx
y
Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
23/35
Solution of homogeneous linear differential
here
19/07/2013 Sci.& Hum.Dept. ,E.M.-3
2e and e .mx mx mx y y me y m
2 20 e e e 0 0.mx mx mx y ay cy m am b m am b
To solve the equation y+ay+by=0 substitute y= emxand try to determine m so thatthis substitution is a solution to the differential equation.
Compute as follows:
Homogeneous linear second order differential equations can always be solved by
certain substitutions.
This follows since emx0 for allx.
The equation m2+ am+ b= 0 is the Characteristic Equation or Auxiliary
equation of the differential equation y + ay + by= 0.
23
A differential equation of the type
y+ay+by=0, a,breal numbers,is a homogeneous linear second order differential
equation with constant coefficients.
23
8/12/2019 Final Lecture 6 Maths-3
24/35
So here we get two roots of Auxiliary eqn. and comparing them with
we get a==1,b= a and c=b and therefore
therefore roots are
Now if then CASE---1 CE m2+am+b=0 has two different real
solutions m1 and m2
1 21 2e e
m x m x y C C
In this case the functions y = em1x and y = em2x are both solutions to the original
given differential equation and the general solution is
Example 0y y CE 2
1 0m 1 or m 1.m
1 2e e
x xy C C General Solution
The fact that all these functions are solutions can be verified by a direct calculation.
3/4/2014 24
2 0ax bx c 2 24 4b ac a b
2 2
1, 2
4 4
2 2
b b ac a a bm m
a
0
Sci.& Hum.Dept. ,E.M.-3 24
8/12/2019 Final Lecture 6 Maths-3
25/35
Solving Homogeneous 2ndOrder Linear
Equations: Case II
CE has real double root m(that means equal
root) is one solution of equation now
Then second sol. And
In this case the functions y = emx and y =
xemx are both solutions to the originalequation and the general sol.is
25
mxy e
2 1y uy
2
1
1 1pdx axax
u e dx e dx xy e
1 2
e ex xy C C x
3/4/2014Sci.& Hum.Dept. ,E.M.-319/07/2013
8/12/2019 Final Lecture 6 Maths-3
26/35
Solving Homogeneous 2ndOrder Linear
Equations: Case III
Now,auxi.eqn has two complex solutions
Now, by Eulers formula
So, our solutions
So , the general sol. is
26
1 2,m m p iq
cos sini
e i
1
2
( )
( )
(cos sin )
(cos sin )
m x p iq x px iqx px
m x p iq x px iqx px
e e e e e qx i qx
e e e e e qx i qx
Sci.& Hum.Dept. ,E.M.-319/07/2013
1 2
1 1
m x m xy c e c e
8/12/2019 Final Lecture 6 Maths-3
27/35
So, General solution in case-3 is
19/07/2013 27
1 2
1 2 1 2
1 2 1 2
1 2
(cos sin ) (cos sin )
[( cos cos ) ( sin sin )]
[( ) cos ( ) sin )]
[ cos sin ]
px px
px
px
px
y c e qx i qx c e qx i qx
e c qx c qx c i qx c i qx
e c c qx c c i qx
e c qx c qx
1 2[ cos sin ]pxy e c qx c qx
Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
28/35
19/07/2013 28Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
29/35
19/07/2013 29Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
30/35
19/07/2013 30Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
31/35
19/07/2013 31Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
32/35
Solution of non- homogeneous linear
diff.eqn.with constant coefficients
19/07/2013 32Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
33/35
The General Solution of non-
homogeneous linear diff.eqn.withconstant coefficients is of the form
Y = Complimentary function +Particular integral
19/07/2013 33Sci.& Hum.Dept. ,E.M.-3
8/12/2019 Final Lecture 6 Maths-3
34/35
8/12/2019 Final Lecture 6 Maths-3
35/35
General method
Shortcut methods
Method of variation Of parameters
Method of undetermined coefficients
19/07/2013 Sci & Hum Dept E M -3 35
Methods of finding Particular integral