Differential Equations Question 9
Question 9 - 2024 (29 Jan Shift 1)
A function $y=f(x)$ satisfies
$f(x) \sin 2 x+\sin x-\left(1+\cos ^{2} x\right) f^{\prime}(x)=0$ with condition $f(0)=0$. Then $f\left(\frac{\pi}{2}\right)$ is equal to
(1) 1
(2) 0
(3) -1
(4) 2
Show Answer
Answer (1)
Solution
$\frac{d y}{d x}-\left(\frac{\sin 2 x}{1+\cos ^{2} x}\right) y=\sin x$
I.F. $=1+\cos ^{2} x$
$y \cdot\left(1+\cos ^{2} x\right)=\int(\sin x) d x$
$=-\cos x+C$
$x=0, C=1$
$y\left(\frac{\pi}{2}\right)=1$
