Differential Equations
Linear ODE in x(y) — Finding mn
nta_pyq_2023_apr
Grade 12
Question:
Let $x=x(y)$, $0<y<\frac{\pi}{2}$, satisfy $(\ln\cos y)^2\cos y\,dx-(1+3x\ln\cos y)\sin y\,dy=0$ with $x(\frac{\pi}{3})=\frac{1}{2\ln 2}$. If $x(\frac{\pi}{6})=\frac{1}{\ln m-\ln n}$, then $mn$ is equal to
Step-by-Step Solution
Key Concept: Rewrite as $\frac{dx}{dy}+\frac{-3\tan y}{\ln\cos y}x=\frac{\sin y}{(\ln\cos y)^2\cos y}$. IF $=(\ln\cos y)^3$.
$x=-\frac{1}{2\ln\cos y}$. $x(\frac{\pi}{6})=\frac{1}{\ln4-\ln3}$. $mn=12$.
Correct Answer: 12