Solve the initial value problem: given that y = 0 when .

Given, A differential equation

To Find: Find the particular solution, when y = 0 and .

Explanation: We have

It can be written as

Now, Divide the equation by tanx , we get

Now, It is a form of the linear differential equation in the form,

When comparing the equation (i) with a linear differential equation, we get

and

Since, the solution of the Linear differential equation is

I.F =

And,

I.F × y

So, The solution for the given linear differential equation is

I.F

I.F

I.F = e^{log sin x}

I.F = sin x

The general solution of this differential equation

Sinx × y

Sinx × y

Sinx × y

Now, Let I₁ =

Integrate I₁ by the Product rule

Now, I₂ will also be solved by the product rule

Here, f(x) = x² and g(x) = cosec x, So

Therefore,

y. sin x = x²sin x + C

y = x² + C

Now, putting y = 0 and .

Now putting the value of C in equation (1)

Hence, This is the particular solution of the given differential equation