Differential Equations
Linear ODE — Integrating Factor with Trig RHS
nta_pyq_2026_jan
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $\sec x\dfrac{dy}{dx}-2y=2+3\sin x$, $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, $y(0)=-\dfrac{7}{4}$. Then $y\!\left(\dfrac{\pi}{6}\right)$ is equal to:
$-\dfrac{5}{4}$
$-\dfrac{5}{2}$
$-3\sqrt{3}-7$
$-3\sqrt{2}-7$
Step-by-Step Solution
Key Concept: Rewrite as $\dfrac{dy}{dx}-2\cos x\cdot y=(2+3\sin x)\cos x$. Integrating factor $\mu(x)=e^{-2\sin x}$. Multiply and integrate both sides.
$y=-\tfrac{7}{4}-\tfrac{3}{2}\sin x$. $y(\pi/6)=-\tfrac{5}{2}$.
Correct Answer: 2