Module 5

Ordinary Differential Equations of Higher Orders

5.1 Linear Differential equation

Let D be symbol which denotes differential with respect to say of function which immediately follows i.e., D stands for

Problem

Solve 

Solution:

Auxiliary equation

Solution is 

5.1.1 Types of auxiliary equation

If roots are a, a, b

Roots

 Roots =

Solution of

where X is a function of X

a)  Definition:   is defined as that function of X which when operated upon by given X.

Particular Integral of equation

5.1.2 Types of P.I

(i) 

(ii) 

(iii)  

                     P.I=

(iv)  

(v)      

(vi)  When any of the above method fails to given P.I

Problem:

Solve

Solution:

Auxiliary equation  

Complementary function

Complete Solution is  

Problem:

Solve

Solution:

Auxiliary equation

C.F is

               []

   The Complete Solution is  

Example

Solve 

Solution:

The Auxiliary equation is

  The C.F is   y=

P.I

   The Complete Solution is 

Example

Solve

Solution:

The Auxiliary equation is   

  The C.F is

P.I

Now,  

   The Complete Solution is

Example

Solve 

Solution:

The auxiliary solution is

  The C.F is

Now,

                  [Putting ]

Similarly, we find that

   The Complete Solution is

Example

Solve

Solution:

The auxiliary equation is

The C.F is

              [Put ]

   The Complete Solution is

Example

Solve

Solution:

The auxiliary equation is     

The C.F is

But

   The Complete Solution is

Example

Solve

Solution:

The auxiliary equation is   

The C.F is

 [By parts]

   The Complete Solution is

Example

Solve

Solution:

The auxiliary equation is    

The C.F is

Here ,   ,  .    Let

Now, 

            Put

 Multiply by in the numerator and denominator

Put  

   The Complete Solution is

Example

Solve

Solution:

The auxiliary equation is   

The C.F is

Now,      

And 

   The Complete Solution

Example

Solve

Solution:

The auxiliary equation is       

The C.F is

And,  

Now, 

And 

        And

    The Complete Solution is

5.2 Method of variations of parameters

We assume

Where

Example:

 Apply variation of parameter to solve   

 Solution:

Auxiliary equation is  

Solving quadratic equation 

C.F 

P.I 

 [Put  ]

Put

 The complete solution is  

5.3 Power series solution :

The general power series solution method.

power series represents a function f on an interval of convergence, and that you can successively differentiate the power series to obtain a series for f , f , and so on.

Example 1 Use a power series to solve the differential equation y - 2y = 0.

Solution     Assume that y =  is a solution. Then, y   = Substituting

for  y’   and  -2y,  you  obtain  the  following  series  form  of  the  differential  equation.

y’  - 2y = 0

 00

anxn  = 0

n =1

       =  2anxn

L (n + 1)an +1xn = L 2anxn

n =0 n =0

Now, by equating coefficients of like terms, you obtain the recursion formula

(n + 1)an +1 = 2an, which implies that

5.4 Legendre's Differential Equation 

Differential Equation which arises in numerous problems, especially in those exhibiting spherical symmetry. Legendre's Differential Equation is defined as:

where  is a real number. The solutions of this equation are called Legendre Functions of degree .

When  is a non-negative integer, i.e., , the Legendre Functions are often referred to as Legendre Polynomials .

Since Legendre's differential equation is a second order ordinary differential equation, two sets of functions are needed to form the general solution. Legendre Polynomials of the second kind  are then introduced. The general solution of a non-negative integer degree Legendre's Differential Equation can hence be expressed as:

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5.4.1 Functions

Rodrigues' Formula: The Legendre Polynomials  can be expressed by Rodrigues' formula

 where 

Generating Function: The generating function of a Legendre Polynomial is

Orthogonality: Legendre Polynomials , , form a complete orthogonal set on the interval . It can be shown that

By using this orthogonality, a piecewise continuous function  in  can be expressed in terms of Legendre Polynomials:

where:

This orthogonal series expansion is also known as a Fourier-Legendre Series expansion or a Generalized Fourier Series expansion.

Even/Odd Functions: Whether a Legendre Polynomial is an even or odd function depends on its degree .

Based on ,

•  is an even function when  is even.

•  is an odd function when  is odd.

In addition, from,

•  is an even function when  is odd.

•  is an odd function when  is even.

Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighbouring Legendre Polynomials at the same point.

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Bessel Differential Equation

The Bessel differential equation is the linear second-order ordinary differential equation given by

Equivalently, dividing through by ,

The solutions to this equation define the Bessel functions  and . The equation has a regular singularity at 0 and an irregular singularity at .

A transformed version of the Bessel differential equation given by Bowman (1958) is

The solution is

where