Unit – 4

Differential equations

4.1 Series solution of differential equations

The general form of linear differential equation of second order is

Where p and q are constants and R is a function of x or constant.

Differential Operator

D stands for operation of differential i.e.

stands for the operator of integration.

stands for operation of integration twice.

Thus,

Note:- Complete solution = complementary function + Particular integral

i.e. y=CF + PI

Method for finding the CF

Step1:- In finding the CF right hand side of the given equation is replaced by zero.

Step 2:- Let be the CF of

Putting the value of in equation (1) we get

It is called auxiliary equation.

Step 3:- Roots Real and Different

If are the roots the CF is

If are the roots then

Step 4- Roots Real and Equal

If both the roots are then CF is

If roots are

Example: Solve

Ans. Given,

Here Auxiliary equation is

Solve:

Or,

Ans. Auxiliary equation are

Note: If roots are in complex form i.e.

Solve:

Ans. Auxiliary equation are

Solve.

Ans. Its auxiliary equation is

Solution is

Solve.

Ans. The auxiliary equation is

Hence the solution is

Rules to find Particular Integral

Case 1:

If,

If,

Solve:

Ans. Given,

Auxiliary equation is

Case2:

Expand by the binomial theorem in ascending powers of D as far as the result of operation on is zero.

Solve.

Given,

For CF,

Auxiliary equation are

For PI

Case 3:

Or,

Solve:

Ans. Auxiliary equation are

Case 4:

Solve.

Ans. AE=

Complete solution is

Solve

Ans. The AE is

Complete solution y= CF + PI

Solve.

Ans. The AE is

Complete solution = CF + PI

Solve.

Ans. The AE is

Complete solutio0n is y= CF + PI

Find the PI of

Ans.

Solve

Ans. Given equation in symbolic form is

Its Auxiliary equation is

Complete solution is y= CF + PI

Solve.

Ans. The AE is

We know,

Complete solution is y= CF + PI

Solve. Find the PI  of  (D2-4D+3)y=ex cos2x

Ans.

Solve. (D3-7D-6) y=e2x (1+x)

Ans. The auxiliary equation i9s

Hence complete solution is y= CF + PI

4.2 Power series method

Working Rule

Step 1 Let be the solution of the given differential equation.

Step 2: Find etc.

Step 3: Substitute the expression of in the given differential equation.

Step 4: Calculate coefficient of various powers of x by equating coefficients to zero.

Step 5: Substitute the values of in the differential equation to get the required series solution.

Example. Solve in series the equation

Ans. 

Since x=0 is the ordinary point of the equation (1)

Then

Substituting in (1) we get

Equating to zero the coefficient of the various powers of x we obtain

Substituting these values in (2) we get

Solve.

Ans. Let,

Substituting the value of   in the given equation we get

Where the first summation extends over all values of K from 2 to

And the second from K =

Now equating the coefficient of equal to zero we have

For K =4

Solve.

Ans. Let

Substituting for in the given differential equation

Equating the coefficients of various powers of x to zero we get

4.3 Legendre equation and Legendre polynomial

Legendre’s equation is

And

Prove that

Ans. We know that

Put n=2

Prove that .

Ans. We know

+

Put x = 1 both sides we get

Equating the coefficient of on both sides we get

Prove that

Ans. We know

Differentiating with respect to z we get

Multiplying both sides by we get

Equating the coefficient of from both sides we get

Solve. Statement

Proof. Let is a solution of

is the solution of

Multiplying (1) by z and (2) by y and subtracting we get

Now integrative -1 to 1 we get

Now we have to prove that

We know that,

Squaring both sides we get

Integrating both sides between -1 to +1 we get

on both sides we get

here n = m

Prove that

Ans. The Recurrence formula is

Pn+1+nPn-1

Replacing n by (n+1) and (n-1) we have

Multiplying (1) and (2) and integrating in the limits -1 to 1 we get

(By orthogonality property)

4.4 Bessel’s function and its properties

The Bessel equation is

Bessel function of first kind

Bessel function of second kind

Recurrence Formula

1)     xJn'=nJn-xJn+1

2)    

3)    

4)    

5)    

6)    

Prove that (1)

Ans. We know

(b) Prove that

Ans. We know that

(3) Prove that

Ans. We know that

Jn(x)=

If n = 0

If n = 1

Note General solution of Bessel Equation

References

1. Ordinary and Partial Differential equations by J. Sihna Ray and S Padhy, Kalyani Publishers

2. Advance Engineering Mathematics by P.V.O’NEIL, CENGAGE

3. Ordinary Differential Equation by P C Biswal , PHI secondedition.

4. Engineering Mathematics by P. S. Das & C. Vijayakumari, Pearson. N.B:Thecourseisof3creditwith4contacthours.