Differential Equations
Linear DE — Integrating Factor with Trig
nta_pyq_2024_jan
Grade 12
Question:
A function $y=f(x)$ satisfies $f(x)\sin2x+\sin x-(1+\cos^2x)f'(x)=0$ with condition $f(0)=0$. Then $f\left(\dfrac{\pi}{2}\right)$ is equal to
Step-by-Step Solution
Key Concept: Rearrange to $f'(x)-\frac{\sin2x}{1+\cos^2x}f(x)=\frac{\sin x}{1+\cos^2x}$. Find I.F. $=1+\cos^2x$. Integrate and apply $f(0)=0$.
$\frac{dy}{dx}-\frac{\sin2x}{1+\cos^2x}y=\frac{\sin x}{1+\cos^2x}$. I.F.$=e^{\int\frac{-\sin2x}{1+\cos^2x}dx}=1+\cos^2x$. $y(1+\cos^2x)=1-\cos x$. $f(\pi/2)=1$.
Correct Answer: 1