Module–2

Ordinary Differential Equations-2

Question Bank

Question-1: Solve (D – D’ – 2 ) (D – D’ – 3) z =

Sol.

The C.F. Will be given by-

Particular integral-

Therefore, the complete solution is-

Question-2: Find the P.I. Of

Sol.

Question-3: Find the P.I. Of (D + 1) (D + D’ – 1)z = sin (x + 2y)

Sol.

Question-4: Find P.I. Of

Sol. P.I =

Replace D by D+1

Put

Question-5: Solve the following DE by using variation of parameters-

Sol. We can write the given equation in symbolic form as-

To find CF-

It’s A.E. Is

So that CF is-     

To find PI-

Here

Now

Thus PI = 

            = 

            = 

             = 

             = 

So that the complete solution is-

Question-6: Solve the following by using the method of variation of parameters.

Sol. This can be written as-

C.F.-

Auxiliary equation is-

So that the C.F. Will be-

P.I.-

Here

Now

Thus PI = 

            = 

            = 

So that the complete solution is-

Question-7: Solve

Sol. As we see that this is a Legendre’s linear equation.

Now put

So that-

And

Then the equation becomes- D (D – 1)y+ Dy + y = 2 sin t

Its auxiliary equation is-

And particular integral-

               P.I. =

Note -

Hence the solution is -

Question-8: Solve

Sol.

Here we have-

Let the solution of the given differential equation be-

Since x = 0 is the ordinary point of the given equation-

Put these values in the given differential equation-

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

Therefore, the solution is-

Question-9: Solve in series the equation-

Sol.

Here we have-

Let us suppose-

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

Then-

And

Put these values in equation (1)-

We get-

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

And so on….

In general, we can write-

Now putting n = 5,

Put n = 6-

Put n = 7,

Put n = 8,

Put n = 9,

Put n = 10,

Put the above values in equation (1), we get-

Question-10: Express in terms of Legendre polynomials.

Sol.

By equating the coefficients of like powers of x, we get-

Put these values in equation (1), we get-

Question-11: Show that-

Sol.

We know that

Equating the coefficients of both sides, we have-

Question-12: Prove that-

Sol.

As we know that-

Now put n = 1/2 in equation (1), then we get-

Hence proved.

Question-13: Prove that-

By using Rodrigue formula for Legendre function.

On integrating by parts, we get-

Now integrating m – 2 times, we get-

Question-14: Prove that-

Sol.

We know that- from recurrence formula

On integrating we get-

On taking n = 2 in (1), we get-

Again-

Put the value of from equation (2) and (3), we get-

By equation (1), when n = 1