# 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 = elog 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

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