Unit-1

Linear differential equations

Question-1: Solve

Sol.

Its auxiliary equation is-

Where-

Therefore the complete solution is-

Questin-2: Solve

Or,

Ans. Auxiliary equation is

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

Question-3: Solve

Ans. Given,

Auxiliary equation is

Question-4:

Given,

For CF,

Auxiliary equation is

For PI

Question-5: Solve

Ans. The AE is

Complete solution y= CF + PI

Question-6: Solve

Ans. The AE is

Complete solutio0n is y= CF + PI

Question-7: solve

Ans. Given equation in symbolic form is

Its Auxiliary equation is

Complete solution is y= CF + PI

Question-8: Find the PI  of(D2-4D+3)y=ex cos2x

Ans.

Question-9: Solve

Solution:

Auxiliary equation  

Complementary function

Complete Solution is  

Question-10: Solve

Solution:

The auxiliary equation is     

The C.F is

But

   The Complete Solution is

Question-11: Solve the following DE by using a 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-12: Solve

Ans. Let,

AE is

y= CF + PI

Question-13: Solve

Sol. As we see that this is 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-14: Solve the following simultaneous differential equations-

Given that x(0)=1  and y(0)= 0

Solution:

Consider the given equations,

Dy +2x = sin2t

Dx -2y = cos2t

By solving the above equations we get,

(D2 +4)Y =0

X(0) = 1, y(0) = 0

A =0, B=-1