Question

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

Answer 1

Answer: 2

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