Let the solution curve x=x(y), 0

Question

Let the solution curve $x=x(y)$, $0<y<\dfrac{\pi}{2}$, of $(\log_e(\cos y))^2\cos y\,dx-(1+3x\log_e(\cos y))\sin y\,dy=0$ satisfy $x\left(\dfrac{\pi}{3}\right)=\dfrac{1}{2\log_e2}$. If $x\left(\dfrac{\pi}{6}\right)=\dfrac{1}{\log_em-\log_en}$, $m,n$ coprime, then $mn$ is \_\_\_\_\_\_.

Answer: 12

Accepted Answer

With $t=\ln\cos y$: $\dfrac{dx}{dy}-\dfrac{3	an y}{\ln\cos y}x = \dfrac{	an y}{(\ln\cos y)^2}$; I.F.$=(\ln\cos y)^{-3}$, giving $x\ln^3(\cos y) = -\dfrac{\ln^2(\cos y)}{2}+C$.

Using $x(\pi/3)=\dfrac{1}{2\ln2}$: $C=0$. At $y=\pi/6$: $x=\dfrac{1}{\ln4-\ln3}$, so $m=4,n=3$, $mn=12$.

Solutions · 1