# Solve the initial value problem 4 (sin (t) dy / dt + cos (t) y) = cos (t) sin^2 (t), for 0 less...

## Question:

Solve the initial value problem {eq}\displaystyle 4 (\sin (t) \frac{dy}{dt} + \cos (t) y) = \cos (t) \sin^2 (t) {/eq}, for {eq}0 < t < \pi {/eq} and {eq}\displaystyle y \bigg(\frac{\pi}{2}\bigg) = 7 {/eq}.

Find the integrating factor, and find y (t).

## Initial Value Problem:

The equation is of the type

{eq}\frac{\mathrm{d} y}{\mathrm{d} t}+Py=Q(f(t)) {/eq} where {eq}e^{Pdt} {/eq} is the integrating factor and the value of the constant can be found using the intial value conditions.

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To solve the equation let us write the general equation

{eq}\frac{\mathrm{d} y}{\mathrm{d} t}+Py=Q(f(t)) {/eq}

Let us rearrange the equation

{eq}\...