Question

Let $\frac{d y}{d x}-2 y \cot x=\cos x$ such that $y\left(\frac{\pi}{2}\right)=0$. If the maximum value of $y$ is $k$, find the value of $k$.

Answer 1

$\text { I.F. }= e ^{-2 \int \cot x}=\frac{1}{\sin ^2 x}$ $\frac{y}{\sin ^2 x}=\int \frac{\cos x}{\sin ^2 x} d x+c=-\operatorname{cosec} x+c $ $y=-\sin x+c \sin ^2 x$ Now, at $x =\frac{\pi}{2}, y =0 \Rightarrow c =1$ $\therefore y=\sin ^2 x-\sin x=\left(\sin x-\frac{1}{2}\right)^2-\frac{1}{4} $ $y_{\max }=\left(-1-\frac{1}{2}\right)^2-\frac{1}{4}=\frac{9}{4}-\frac{1}{4}=2 $