Solve the differential equation: , given that y = 0 when x = 1.

OR

Solve the differential equation: , subject to the initial condition y(0) = 0.

Given equation:

On-rearranging the term, we get,

Now, this is a homogenous differential equation of order 1.

Let y = vx and

Therefore,

Integrating both sides, we get,

Now, we know that,

and

Therefore,

Putting the value of y, we get,

At x = 1, y = 0,

C = 1

Hence,

OR

Given Equation:

Dividing the whole equation by (1 + x²), we get,

Now, this is a linear equation of the form,

We know that the solution of this equation is given by,

Where

Therefore, for a given equation,

Let 1 + x² = t

Differentiating both sides, we get,

2x dx = dt

Therefore,

The solution of the equation:

At x = 0, y = 0

Therefore,

Hence,