JEE PYQ: Differential Equation Question 2
Question 2 - 2021 (16 Mar Shift 2)
If $y = y(x)$ is the solution of the differential equation $\frac{dy}{dx} + (\tan x)y = \sin x$, $0 \le x \le \frac{\pi}{3}$, with $y(0) = 0$, then $y\left(\frac{\pi}{4}\right)$ is equal to:
(1) $\frac{1}{4}\log_e 2$
(2) $\left(\frac{1}{2\sqrt{2}}\right)\log_e 2$
(3) $\log_e 2$
(4) $\frac{1}{2}\log_e 2$
Show Answer
Answer: (2)
Solution
I.F. $= e^{\int \tan x,dx} = \sec x$. Solution: $y\sec x = \int \tan x,dx = \ln|\sec x| + C$. At $x = 0$, $y = 0$: $C = 0$. So $y\sec x = \ln|\sec x|$, i.e. $y = \cos x\cdot\ln|\sec x|$. At $x = \frac{\pi}{4}$: $y = \frac{1}{\sqrt{2}}\cdot\ln\sqrt{2} = \frac{1}{2\sqrt{2}}\log_e 2$.
BETA
