S1) Here This implies that .

Differentiating, we get

     .

     .

    .

The   Taylor’s series at ,

  (1)

At in equation (1) we get

At in equation (1) we get

S2) Here .

We have

Differentiating, we get

  implies that or

  implies that or .

   implies that

  implies that

The   Taylor’s series at ,

 Or

Here

The   Taylor’s series

       .

S3) Given equation

Here

Also

By  Runge Kutta formula for first  interval

Again

A fourth  order Runge Kutta formula:

To  find y at 

A fourth  order Runge Kutta formula:

S4) Given equation 

Here 

Also

By  Runge Kutta formula for first  interval

         )

 A fourth  order Runge Kutta formula:

Hence at x  = 0.2 then y  = 1.196

To find the  value of y at x=0.4. In this case

 A fourth  order Runge Kutta formula:

Hence at  x  = 0.4 then  y=1.37527

S5) Given second order differential equation is

Let   then above equation reduces to

     Or

(say)

Or .

By RungeKutta Method we have

A fourth  orderRungeKutta formula:

Q6) Solve the differential equations

for

Using four order RungeKutta method with initial conditions

S6)Given differential equation are

Let

And

Also

 By RungeKutta Method we have

A fourth order RungeKutta formula:

And  

.