Unit – 3

Numerical solutions of Ordinary Differential Equations

Q1) What is fourth order Runge-kutta method?

A1)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite  as 

A fourth  order Runge Kutta formula:

Where 

Q2) Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

A2)

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:

Q2) Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

A3)

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Q3) Using Runge Kutta method of fourth order,  solve

A4)

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

Q4) What is Picard’s method?

A5)

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Q5) Find the value of y for x = 0.1 by using Picard’s method, given that-

A6)

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Q6) Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

A7)

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Q7) Using Runge Kutta method of order four, solve   to find

A8)

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Q8) Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

A9)

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Q9) Solve the differential equations

       for

A10)

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Unit – 3

Numerical solutions of Ordinary Differential Equations

Q1) What is fourth order Runge-kutta method?

A1)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite  as 

A fourth  order Runge Kutta formula:

Where 

Q2) Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

A2)

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:

Q2) Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

A3)

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Q3) Using Runge Kutta method of fourth order,  solve

A4)

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

Q4) What is Picard’s method?

A5)

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Q5) Find the value of y for x = 0.1 by using Picard’s method, given that-

A6)

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Q6) Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

A7)

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Q7) Using Runge Kutta method of order four, solve   to find

A8)

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Q8) Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

A9)

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Q9) Solve the differential equations

       for

A10)

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Unit – 3

Unit – 3

Unit – 3

Numerical solutions of Ordinary Differential Equations

Q1) What is fourth order Runge-kutta method?

A1)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite  as 

A fourth  order Runge Kutta formula:

Where 

Q2) Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

A2)

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:

Q2) Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

A3)

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Q3) Using Runge Kutta method of fourth order,  solve

A4)

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

Q4) What is Picard’s method?

A5)

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Q5) Find the value of y for x = 0.1 by using Picard’s method, given that-

A6)

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Q6) Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

A7)

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Q7) Using Runge Kutta method of order four, solve   to find

A8)

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Q8) Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

A9)

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Q9) Solve the differential equations

       for

A10)

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .

Unit – 3

Unit – 3

Numerical solutions of Ordinary Differential Equations

Q1) What is fourth order Runge-kutta method?

A1)

Consider the differential equation of first order

Let  be the first interval.

A second order Runge Kutta formula  

Where

Rewrite  as 

A fourth  order Runge Kutta formula:

Where 

Q2) Use Runge Kutta method to find y when x=1.2  in step of h=0.1 given that

A2)

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:

Q2) Apply Runge Kutta fourth order method to find an approximate value of  y for  x=0.2   in step of 0.1,  if

A3)

Given  equation 

Here 

Also

By  Runge Kutta formula for first  interval

A fourth  order Runge Kutta formula:

Again

A fourth  order Runge Kutta formula:

Q3) Using Runge Kutta method of fourth order,  solve

A4)

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

Q4) What is Picard’s method?

A5)

Picard’s method-

Let us suppose the first order equation-

It is required to find out that particular solution of equation (1) which assumes the value when ,

Now integrate (1) between limits, we get-

This is equivalent to equation (1),

For it contains the not-known y under the integral sign,

As a first approximation to the solution, put in f(x, y) and integrate (2),

For second approximation-

Similarly-

And so on.

Q5) Find the value of y for x = 0.1 by using Picard’s method, given that-

A6)

We have-

For first approximation, we put y = 1, then-

Second approximation-

We find it very hard to integrate.

Hence we use the first approximation and take x = 0.1 in (1)

Q6) Obtain the picard’s second approximation for the given initial value problem-

Find y(1).

A7)

The first approximation will be-

Replace y by , we get-

The second approximation is-

The third approximation-

It is very difficult to solve the integration-

This is the disadvantage  of the method.

Now we get from the second approximation-

At x = 1-

Q7) Using Runge Kutta method of order four, solve   to find

A8)

Given second order differential equation is

Let   then above equation reduces to

     Or

   (say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

Q8) Using Runge Kutta method, solve

   for correct to four decimal places with initial condition .

A9)

Given second order differential equation is

Let   then above equation reduces to

Or

(say)

Or .

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And

.

Q9) Solve the differential equations

       for

A10)

Using four order Runge Kutta method with initial conditions

Given differential equation are

Let

And

Also

By Runge Kutta Method we have

A fourth  order Runge Kutta formula:

And  

 .