Differential Equations
Linear ODE — Quotient Rule Form
nta_pyq_2026_jan
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $x\dfrac{dy}{dx}-y=x^2\cot x$, $x\in(0,\pi)$. If $y\!\left(\dfrac{\pi}{2}\right)=\dfrac{\pi}{2}$, then $6y\!\left(\dfrac{\pi}{6}\right)-8y\!\left(\dfrac{\pi}{4}\right)$ is equal to:
Step-by-Step Solution
Key Concept: Rewrite as $\dfrac{dy}{dx}-\dfrac{y}{x}=x\cot x$. IF $=\tfrac{1}{x}$. Gives $\dfrac{d}{dx}\!\left(\tfrac{y}{x}\right)=\cot x$. Integrate: $\tfrac{y}{x}=\ln|\sin x|+C$.
$y=x(1+\ln\sin x)$. $6y(\pi/6)-8y(\pi/4)=-\pi$.
Correct Answer: 3