JEE PYQ: Differential Equation Question 4
Question 4 - 2021 (17 Mar Shift 2)
Let $y = y(x)$ be the solution of the differential equation $\cos x(3\sin x + \cos x + 3),dy = (1 + y\sin x(3\sin x + \cos x + 3)),dx$, $0 \le x \le \frac{\pi}{2}$, $y(0) = 0$. Then $y\left(\frac{\pi}{3}\right)$ is equal to:
(1) $2\log_e\left(\frac{2\sqrt{3}+9}{6}\right)$
(2) $2\log_e\left(\frac{2\sqrt{3}+10}{11}\right)$
(3) $2\log_e\left(\frac{\sqrt{3}+7}{2}\right)$
(4) $2\log_e\left(\frac{3\sqrt{3}-8}{4}\right)$
Show Answer
Answer: (2)
Solution
Rewriting: $\frac{dy}{dx} - (\tan x)y = \frac{1}{\cos x(3\sin x + \cos x + 3)}$. I.F. $= \cos x$. Solution: $y\cos x = \int \frac{dx}{3\sin x + \cos x + 3}$. Using $\tan\frac{x}{2} = t$: $I = \int \frac{\sec^2\frac{x}{2},dx}{2(\tan^2\frac{x}{2} + 3\tan\frac{x}{2} + 2)} = \ln\left|\frac{\tan\frac{x}{2}+1}{\tan\frac{x}{2}+2}\right| + C$. With $y(0) = 0$: $C = \ln 2$. At $x = \frac{\pi}{3}$: $y = 2\log_e\left(\frac{2\sqrt{3}+10}{11}\right)$.
BETA
