Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& \frac{dy}{dx}=\frac{7\cot y}{x}-\frac{e^x\mathrm{cosec}y}{x^5}\\ & \frac{dy}{dx}=\frac{7\cot y}{\sin y\cdot x}-\frac{e^x}{\sin y{x}^5}\\ & \sin y\frac{dy}{dx}-\cos y\cdot \frac{7}{x}=\frac{-e^x}{x^5}\\ & \text{ let }-\cos y=t\\ & \sin y\frac{dy}{dx}=\frac{dt}{dx}\end{array}$
$\begin{array}{rl}& \frac{dt}{dx}+\frac{7t}{x}=\frac{-e^x}{x^5}\\ & \text{ I.F. }=x^7\\ & \text{ t. }{x}^7=-\int x^2{e}^{x}dx\\ & \mathrm{cosy}{x}^7=x^2{e}^x-2\int x{e}^{x}dx\\ & \mathrm{cosy}{x}^7=x^2{e}^x-2x{e}^x+2e^x+c\\ & x=1,y=\frac{\pi }{2},c=-e\\ & \mathrm{cosy}=\frac{2e^2-e}{128}\\ & \text{ option }(3)\end{array}$