Unit 1

Linear Differential Equations (LDE) and Applications

Q (1) Solve sin2x= y+tanx

Ans-

Soln :-

The given equation can be written as,

which is a linear diff. eqn.

Now

Thus the solution is.

=

Which is the required solution.

Q(2) Solve  +

Ans-

Soln :-

 This is a linear differential equation

Considr

Hence it’s solution is,

Q(3) Solve

Soln :-

The given eqn can be written as,

which is a linear diff eqn of the form

Now

Hence  it’s solution  is

Thus 

Is the required solution of given du=ifferential equation.

Q(4) Solve(1 + x + xy2)dy + (y + y3)dx = 0

Ans-

Soln :-

The given eqn can be written as,

which is of the form,

Now,

Hence solution is,

Q(5)  solve

Ans-

Solution:-

     for C.F

      ( put

Hence the complete solution is

Q(6)   solve

Ans-

General solution is

Q(7) solve

Ans-

We have ,

The general solution is

Q(8) solve

Ans-

Q(9) Apply method of variation of parameter to solve

Ans-

Given equation is   

 Or

  The auxiliary equation is

Or

Then

Where are arbitrary constants.

Let P.I =

Where

Now ,

Also

   P.I =

    P.I 

Hence the general solution is

Is the required solution of the given equation.

Q(10) Apply method of variation of parameter to solve

Ans-

Given equation 

  The auxiliary equation is

Or

Then

Where are arbitrary constants.

Let P.I =

Where

     Now ,

Also

 We have P.I =

On substitution we get

    P.I. = 

Hence the general solution is

This is the required solution of the given equation.

Q(11) Use the method of variation to solve

Ans-

Given equation is 

 Or

Therefore the auxiliary equation is

Or

Then C.F.

Where are arbitrary constant.

Now, P.I =

Where

     Now ,

 Also

We have P.I =

On substitution we get

    P.I. = 

Hence the general solution is

This is the required solution of the given equation.

Q(12) Solve

Ans-

Given equation is 

Let

Assume

Substituting in above equation we get

  The auxiliary equation is

Or

Then solution is

Or

This is the required solution of the given equation.

Q(13) Solve

Ans-

Given equation is 

 Let

Assume

Substituting in above equation we get

  +

  The auxiliary equation is

Or

Then C.F.

= 

P.I.

Hence the general solution is

This is the solution of the given equation.

Q(14) Solve

Ans-

Given equation can be re-write as

Let

Assume

Substituting in above equation we get

Or

Or 

The auxiliary equation is

Or

Then C.F.  

Or C.F. =

P.I. = 

     (

   Hence the general solution is

This is the required solution of  the  given equation.

Q(15) Solve

Ans-

Given equation can re-write

  {

Let

 Assume

Substituting in above equation we get

{(logz)

The auxiliary equation is

Or

Then C.F.  

Or C.F.

 Now, P.I.

        Hence the general solution is

           +

This is the required solution of the given equation.

Q(16): Solve 

Ans-

Given equation is

 Let

Assume

Substituting in above equation we get

The auxiliary equation is

Or 2,3

Then C.F.  

Or C.F.

Now, P.I. =

              =

Hence the general solution is

This  is the required solution of the given equation.

Que(17) solve

Ans-

Writing in terms of operator   we have

----1

Solving for x (i.e eliminating y  )

Operating 1 by (D+2 ) we have ,

Or ---3

Multiplying eq 2 by 3 we get

Adding 3 and 4 we get

This is a linear differential equation with constant coeff.

Hence the general solution for x is

Next general solution for (y)

Differentiating equation 6 with respect to t

Putting value of x and dx/dt in equation 1  we get

Simplifying we get

Hence equation 6 and 7 together are the general  solutions

Q(18) solve

Ans-

Consider

Or

By integrating both sides

—1

Which is the first solution

Now consider

Cancelling the common factor , we have

Integrate both sides we get

Equation 1 and 2 taken together constitute the answer