Question

Let \(y=y(x)\) be the solution of the differential equation \(x\,dy=(y+x^3\cos x)\,dx\) with \(y(\pi)=0\), then \(y\left(\frac\pi2\right)\) is equal to :

• A. \(\frac{\pi^2}{4}+\frac\pi2\)
• B. \(\frac{\pi^2}{2}+\frac\pi4\)
• C. \(\frac{\pi^2}{2}-\frac\pi4\)
• D. \(\frac{\pi^4}{4}-\frac\pi2\)

Hint:

Rewrite as \(\frac{dy}{dx}-\frac yx=x^2\cos x\) with integrating factor \(\frac1x\).

Answer 1

The Correct Option is A

\(\frac{dy}{dx}-\frac{y}{x}=x^2\cos x\). Integrating factor \(=e^{-\int dx/x}=\frac1x\).
\(\frac{d}{dx}\left(\frac yx\right)=x\cos x\Rightarrow\frac yx=x\sin x+\cos x+C\).
\(y(\pi)=0\): \(0=0-1+C\Rightarrow C=1\). So \(y=x^2\sin x+x\cos x+x\).
\(y\left(\frac\pi2\right)=\frac{\pi^2}{4}(1)+0+\frac\pi2=\frac{\pi^2}{4}+\frac\pi2\).
Answer: (1) \(\frac{\pi^2}{4}+\frac\pi2\)