Differential Equations
Exact DE — Grouping as Exact Differential
nta_pyq_2026_jan
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $x^4\,dy+\left(4x^3y+2\sin x\right)dx=0$, $x>0$, $y\!\left(\dfrac{\pi}{2}\right)=0$. Then $\pi^4 y\!\left(\dfrac{\pi}{3}\right)$ is equal to:
Step-by-Step Solution
Key Concept: Rewrite: $d(x^4 y)+2\sin x\,dx=0$. Integrating: $x^4 y-2\cos x=C$. Apply $y(\pi/2)=0$: $C=0$. So $x^4 y=2\cos x$.
$x^4 y=2\cos x$. $\pi^4 y(\pi/3)=81$.
Correct Answer: 4