Optionals IAS Mains Mathematics: Questions 124 - 129 of 283

Given differential Equation

Let

Diff. (1) w. r. t

Using (1) and (2)

But,

(3) reduce to,

Integrating,

Where c is the arbitrary constant

Given,

S. F is

Taking Laplace transform of this equation, we get

Putting and we get

Taking inverse Laplace Transform. We get,

Now,

And

Suppose >

is given power series and

Let R be the radius of convergence of the power series then R is also radius of convergence of the power series which is the derived series of

Let z be any point within the circle

Also there exist a real number such that

Then the series is convergent

For

is bounded

So, there exist a finite real number such tha…

If is the distance covered in t (min) , then

At, the velocity of train

The system of Equation is,

Given, Equation can be written as

According to the Gauss – Seidel Method, the approximation of the unknown is given by

Let us take the initial approximation to be zero for each unknown

Iteration (1)

Iteration (2)

Iteration (3)

Iteration (4)

Hence, after 4 iteration the solution of given system of equation is

Given, ordinary differential equation is

Let

Differentiating (1) with respect to we get

From (1)

Integrating, we get

Where is the arbitrary constant

Where is the

Where

Using (1) ,

Putting in (4) , we get

So, required solution is