Differential Equations
Linear DE — Integrating Factor with Trig
nta_pyq_2024_jan
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}=\dfrac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)}$, $x\in\left(0,\dfrac{\pi}{2}\right)$ satisfying the condition $y\left(\dfrac{\pi}{4}\right)=2$. Then $y\left(\dfrac{\pi}{3}\right)$ is
$\sqrt{3}\left(2+\log_e\sqrt{3}\right)$
$\dfrac{\sqrt{3}}{2}(2+\log_e3)$
$\sqrt{3}\left(1+2\log_e3\right)$
$\sqrt{3}\left(2+\log_e3\right)$
Step-by-Step Solution
Key Concept: Simplify the denominator: $\sin x(\sec x-\sin x\tan x)=\sin x\cdot\frac{1-\sin^2x}{\cos x}=\sin x\cos x=\frac{\sin2x}{2}$... leading to a linear DE. I.F. involves $|\tan x|$.
Simplify DE: $\frac{dy}{dx}-2\csc(2x)\cdot y=\sec^2x$. I.F.$=\frac{1}{|\tan x|}$. Integrate: $\frac{y}{|\tan x|}=\ln|\tan x|+c$. IC: $c=2$. $y=|\tan x|(\ln|\tan x|+2)$. $y(\pi/3)=\sqrt{3}(\ln\sqrt{3}+2)=\sqrt{3}(2+\log_e\sqrt{3})$.
Correct Answer: 1