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 
.
