Unit – 6

Ordinary differential equations of higher order

Q1) Solve the following DE by using variation of parameters-

A1) 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-

Q2) Solve the following by using the method of variation of parameters.

A2) 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-

Q3) Solve

A3) As it is a Cauchy’s homogeneous linear equation.

Put

Then the equation becomes

[D(D-1)-D+1] y = t   or

Auxiliary equation-

So that-

C.F.=

Hence the solution is- , we get-

Q4) Solve

A4) 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 -

Q5) Solve

A5) 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-

Q6) Express in terms of Legendre polynomials.

A6) 

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

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

Q7) Show that-

A7) We know that

Equating the coefficients of both sides, we have-

Q8) Prove that-

A8)

By using Rodrigue formula for Legendre function.

On integrating by parts, we get-

Now integrating m – 2 times, we get-

Q9) Prove that-

A9)

Put n = -1/2 in equation (1) of the above question, we get-

Q10) Prove that-

A10)

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