Differential Equations
Linear ODE — Integrating Factor and Verification
nta_pyq_2023_apr
Grade 12
Question:
Let $y=y(x)$ be a solution of $(x\cos x)\,dy+(xy\sin x+y\cos x-1)\,dx=0$, $0<x<\dfrac{\pi}{2}$. If $\dfrac{\pi}{3}y\!\left(\dfrac{\pi}{3}\right)=\sqrt{3}$, then $\left|\dfrac{\pi}{6}y''\!\left(\dfrac{\pi}{6}\right)+2y'\!\left(\dfrac{\pi}{6}\right)\right|$ is equal to _______.
Step-by-Step Solution
Key Concept: Rewrite as $\frac{dy}{dx}+y\frac{x\sin x+\cos x}{x\cos x}=\frac{1}{x\cos x}$. IF $=e^{\int(\tan x+1/x)dx}=x\sec x$. Solution: $xy\sec x=\tan x+C$.
$xy''(x)+2y'(x)=-2\sin(x+\frac{\pi}{3})$. At $x=\frac{\pi}{6}$: value $=-2$. $|\cdot|=2$.
Correct Answer: 2