Solution of Differential Equations

by

Pragyansri Pathi

Graduate student

Department of Chemical Engineering,

FAMU-FSU College of Engineering,FL-32310

Topics to be discussed

Solution of Differential equations

Power series method

Bessel’s equation by Frobenius method

POWER SERIES METHOD

Power series method

Method for solving linear differential equations with variable co-efficient.

)()()( ''' xfyxqyxpy

p(x) and q(x) are variable co-effecients.

Power series method (Cont’d)

Solution is expressed in the form of a power series.

Let ..........33

2210

0

xaxaxaaxaym

mm

so, ..........32 2321

1

1'

xaxaaxmaym

mm

..........3.42.32)1( 2432

2

2'' xaxaaxammym

mm

Power series method (Cont’d)

Substitute ''' ,, yyy

Collect like powers of x

Equate the sum of the co-efficients of each occurring power of x to zero.

Hence the unknown co-efficients can be determined.

Examples

Example 1: Solve 0' yy

Example 2: Solve xyy 2'

Example 3: Solve 0'' yy

Power series method (Cont’d)

• The general representation of the power series solution is,

......)()()( 202010

00

xxaxxaaxxaym

mm

Theory of power series method

Basic concepts

A power series is an infinite series of the form

(1) ......)()()( 202010

00

xxaxxaaxxam

mm

x is the variable, the center x0, and the coefficients a0,a1,a2 are real.

Theory of power series method (Cont’d)

(2) The nth partial sum of (1) is given as,

nnn xxaxxaxxaaxs )(......)()()( 0

202010

where n=0,1,2..

(3) The remainder of (1) is given as,

......)()()( 202

101

n

nn

nn xxaxxaxR

Convergence Interval. Radius of convergence

• Theorem: Let ......)()()( 202010

00

xxaxxaaxxaym

mm

be a power series ,then there exists some number ∞≤R≤0 ,called its radius of convergence such that the series is convergent for

Rxx )( 0 Rxx )( 0and divergent for

• The values of x for which the series converges, form an interval, called the convergence interval

Convergence Interval. Radius of convergence

What is R?

The number R is called the radius of convergence. It can be obtained as,

m

m

m a

aR

1lim

1

or m

mm

aR

lim

1

Examples (Calculation of R)

Example 1: 12n

nnx

n

xn

xn

xn

termn

termnn

n

n

n

.2

)1(

)(

2

2

)1(

)(

)1( 1

2

1

Answer

As n→∞,2.2

)1( x

n

xn

Examples (Calculation of R) Cont’d

Hence 2R

• The given series converges for

2)0( x

• The given series diverges for

2)0( x

Examples

• Example 2:

Find the radius of convergence of the following series,

0 2.nn

n

n

x

Examples

• Example 3:

Find the radius of convergence of the following series,

0 1)...3).(2).(1.(n

n

nnnn

x

Operations of Power series: Theorems

,)()(00

n

nan

n

nan xxbxxa

(1) Equality of power series

If with 0R

then nn ba ,for all n

Corollary

If

0

,0)(n

an xxa all an=0, for all n ,R>0

Theorems (Cont’d)

0n

nnxay

(2) Termwise Differentiation

If is convergent, then

derivatives involving y(x) such as

are also convergent.

If

0n

nnxa and

0n

nnxb are convergent

in the same domain x, then the sum also converges in that domain.

etcxyxy ),(),( '''

(3) Termwise Addition

Existence of Power Series Solutions. Real Analytic Functions

• The power series solution will exist and be unique provided that the variable co-efficients p(x), q(x), and f(x) are analytic in the domain of interest.

What is a real analytic function ?

A real function f(x) is called analytic at a point x=x0 if it can be represented by a power series in powers of x-x0 with radius of convergence R>0.

Example

• Let’s try this

0)()()( ''' xyxxyxy

BESSEL’S EQUATION.

BESSEL FUNCTIONS Jν(x)

Application

• Heat conduction • Fluid flow• Vibrations• Electric fields

Bessel’s equation

• Bessel’s differential equation is written as

0)( 22'''2 ynxxyyx

or in standard form,

0)1(1

2

2''' y

x

nyx

y

Bessel’s equation (Cont’d)

• n is a non-negative real number.• x=0 is a regular singular point.• The Bessel’s equation is of the type,

)(

)(

)(

)(

)(

)( '''

xr

xfy

xr

xqy

xr

xpy

and is solved by the Frobenius method

Non-Analytic co-efficients –Methods of Frobenius

• If x is not analytic, it is a singular point.

)()()()( ''' xfyxqyxpyxr

→ )(

)(

)(

)(

)(

)( '''

xr

xfy

xr

xqy

xr

xpy

The points where r(x)=0 are called as singular points.

Non-Analytic co-efficients –Methods of Frobenius (Cont’d)

• The solution for such an ODE is given as,

0m

mm

r xaxy

Substituting in the ODE for values of y(x),

)(),( ''' xyxy , equating the co-efficient of xm and obtaining the roots gives the indical solution

Non-Analytic co-efficients –Methods of Frobenius (Cont’d)

We solve the Bessel’s equation by Frobenius method.

Substituting a series of the form,

0m

rmmxay

Indical solution

)0(1 nnr nr 2

General solution of Bessel’s equation

• Bessel’s function of the first kind of order n

is given as,

0

2

2

)!(!2

)1()(

mnm

mmn

n mnm

xxxJ

Substituting –n in place of n, we get

02

2

)!1(!2

)1()(

mnm

mmn

n mnm

xxxJ

Example

• Compute

)(0 xJ

• Compute

)(1 xJ

General solution of Bessel’s equation

• If n is not an integer, a general solution of Bessel’s equation for all x≠0 is given as

)(.)(.)( 21 xJcxJcxy nn

Properties of Bessel’s Function

J0(0) = 1

Jn(x) = 0 (n>0)

J-n(x) = (-1)n Jn(x)

)()( 1 xJxxJxdx

dn

nn

n

)()( 1 xJxxJxdx

dn

nn

n

Properties of Bessel’s Function (Cont’d)

)()(2

1)( 11 xJxJxJ

dx

dnnn

)(.)(..2)(. 11 xJxxJnxJx nnn

CxJxdxxJx nn

n

n )())((1

CxJxdxxJx nn

n

n

)())((1

Properties of Bessel’s Function (Cont’d)

)(.2)()( '11 xJxJxJ nnn

)(.2

)()( 11 xJx

nxJxJ nnn

xx

xJ sin2

)(2

1

xx

xJ cos2

)(2

1

Examples

• Example : Using the properties of Bessel’s functions compute,

)(3 xJ1.

2. dxxJx ))((2

2

3. )(2

3 xJ