JEE Main 2025 — 4 April, Evening Shift — Question 24

$\left(7 x^{4} \cot y-e^{x} \operatorname{cosec} y\right) \frac{d x}{d y}=x^{5}$

$x^{5} \frac{d y}{d x}-7 x^{4} \cot y=-e^{x} \operatorname{cosec} y$

$\frac{d y}{d x}-\frac{7}{x} \cot y=-\frac{e^{x}}{x^{5}} \operatorname{cosec} y$

$\sin y \frac{d y}{d x}-\frac{7}{x} \cos y=-\frac{e^{x}}{x^{5}}$

Let $-\cos y=t$ $\sin y \frac{d y}{d x}=\frac{d t}{d x}$

$\therefore \frac{d t}{d x}+\frac{7}{x} t=-\frac{e^{x}}{x^{5}}$

$\therefore$ I.F. $=e^{\int \frac{7}{x} d x}=x^{7}$

$t \cdot x^{7}=\int \frac{-e^{x}}{x^{5}} \cdot x^{7} d x$

$-\cos y \cdot x^{7}=-\int e^{x} x^{2} d x$

$\cos y x^{7}=e^{x}\left(x^{2}-2 x+2\right)+c$

$\because \quad x=1$ then $y=\frac{\pi}{2} \Rightarrow c=-e$

$\therefore \quad \cos y \cdot x^{7}=e^{x}\left(x^{2}-2 x+2\right)-e$

When $x=2$ then $\cos y=\frac{2 e^{2}-e}{128}$