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

Question

Let the solution curve $x=x(y),0<y<\frac{\pi }{2}$, of the differential equation ${({\log }_{e}(\cos y))}^{2}\cos ydx-(1+3x{\log }_{e}(\cos y))\sin ydy=0$ satisfy $x(\frac{\pi }{3})=\frac{1}{2{\log }_{e}2}$. If $x(\frac{\pi }{6})=\frac{1}{{\log }_{e}m-{\log }_{e}n}$, where $m$ and $n$ are coprime, then $mn$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 12

Accepted Answer

$\begin{gathered} (\cos y)(\ln (\cos y){)}^{2}dx=(1+3x\ln \cos y)\sin ydy \ \Rightarrow \frac{dx}{dy}=\frac{(1+3x\ln \cos y)\sin y}{(\ln \cos y{)}^{2}\cos y} \ =	an y[\frac{1}{(\ln \cos y{)}^{2}}+\frac{3x}{\ln \cos y}] \ \Rightarrow \frac{dx}{dy}-(\frac{3	an y}{\ln \cos y})x=\frac{	an y}{(\ln \cos y{)}^{2}} \end{gathered}$

Which is a linear differential equation.

$={e}^{-\int \frac{3	an y}{\ln \cos y}dy}=(\ln \cos y{)}^{3}$

So, the solution is :

$\begin{gathered} x	imes (\ln \cos y{)}^{3}=\int ((\ln \cos y{)}^{3}	imes \frac{	an y}{(\ln \cos y{)}^{2}})dy \ x	imes (\ln \cos y{)}^{3}=\frac{-(\ln \cos y{)}^{2}}{2}+C \end{gathered}$

$y=\frac{\pi }{3}$

$\begin{gathered} \frac{1}{2\ln 2}	imes {(\ln (\frac{1}{2}))}^{3}=-\frac{{(\ln (\frac{1}{2}))}^{2}}{2}+C \ \Rightarrow C=0 \end{gathered}$

$\begin{gathered} x	imes {\ln }^{3}\cos y=\frac{-{\ln }^{2}\cos y}{2} \ y=\frac{\pi }{6},x	imes {(\ln (\frac{\sqrt{3}}{2}))}^{3}=-\frac{1}{2}{(\ln (\frac{\sqrt{3}}{2}))}^{2} \ \Rightarrow x=-\frac{1}{2\ln (\frac{\sqrt{3}}{2})} \end{gathered}$

$\begin{gathered} =-\frac{1}{2[\ln \sqrt{3}-\ln 2]}=\frac{-1}{2[\frac{1}{2}\ln 3-\ln 2]} \ =\frac{-1}{2[\frac{\ln 3-\ln 4}{2}]}=\frac{1}{\ln 4-\ln 3} \ \Rightarrow m=4,n=3 \ \Rightarrow mn=12 \end{gathered}$

Solutions · 1