Let $y=y(x)$ be the solution of the differential equation ${x}^{4}dy+\left(4{x}^{3}y+2\sin x\right)dx=0$, $x0,y\left(\frac{\pi }{2}\right)=0$. Then ${\pi }^{4}y\left(\frac{\pi }{3}\right)$ is equal to: [JEE Main 2026, 23 Jan (Shift 2)]

Question

Let $y=y(x)$ be the solution of the differential equation ${x}^{4}dy+\left(4{x}^{3}y+2\sin x\right)dx=0$, $x>0,y\left(\frac{\pi }{2}\right)=0$. Then ${\pi }^{4}y\left(\frac{\pi }{3}\right)$ is equal to: [JEE Main 2026, 23 Jan (Shift 2)]

YESPYQ · Accepted Answer

Correct answer: c) 81. Given: $\left(x^4 d y+4 x^3 y d x\right)=-2 \sin x d x$ $\Rightarrow \int d\left(x^4 y\right)=\int-2 \sin x d x$ $\Rightarrow x^4 y=2 \cos x+c$ As $y\left(\frac{\pi}{2}\right)=0$ So, $c=0$ Now, $\left(\frac{\pi}{3}\right)^4 y\left(\frac{\pi}{3}\right)=2 \cos \frac{\pi}{3}$ $\Rightarrow \pi^4 y\left(\frac{\pi}{3}\right)=81$

Let \(y=y(x)\) be the solution of the differential equation \({x}^{4}dy+\left(4{x}^{3}y+2\sin x\right)dx=0\), \(x>0,y\le…

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