Let y = y(x) be the solution of the differential equation : x( _e( x))^2 d y+( x-3 y x _e(…

Question

Let y = y(x) be the solution of the differential equation :

$\cos x\left(\log _e(\cos x)\right)^2 d y+\left(\sin x-3 y \sin x \log _e(\cos x)\right) d x=0$, x \in  (0, $\frac{\pi}{2}$ ). If $ y(\frac{\pi}{4}) $ = $-\frac{1}{\log_{e}2}$, then $ y(\frac{\pi}{6}) $ is equal to :

Accepted Answer

$$\begin{aligned}
& \cos x(\ln (\cos x))^2 d y+(\sin x-3 y(\sin x) \ln (\cos x)) d x=0 \
& \cos x(\ln (\cos x))^2 \frac{d y}{d x}-3 \sin x \cdot \ln (\cos x) y=-\sin x \
& \frac{d y}{d x}-\frac{3 \tan x}{\ln (\cos x)} y=\frac{-\tan x}{(\ln (\cos x))^2} \
& \frac{d y}{d x}+\frac{3 \tan x}{\ln (\sec x)} y=\frac{-\tan x}{(\ln (\sec x))^2} \
& \text { I.F. }=e^{\int \frac{3 \tan x}{\ln (\sec x)} d x}=(\ln (\sec x))^3
\end{aligned}$$

$$\begin{aligned}
& y \times(\ln (\sec x))^3=-\int \frac{\tan x}{(\ln (\sec x))^2}(\ln (\sec x))^3 d x \
& y \times(\ln (\sec x))^3=-\frac{1}{2}(\ln (\sec x))^2+C \
& \text { Given }: x=\frac{\pi}{4}, y=-\frac{1}{\ln 2} \
& \frac{-1}{\ln 2} \times(\ln \sqrt{2})^3=-\frac{1}{2} \times(\ln \sqrt{2})^2+C \
& \Rightarrow \frac{-1}{8 \ln 2} \times(\ln 2)^3=\frac{-1}{2} \times \frac{1}{4}(\ln 2)^2+C \
& -\frac{1}{8}(\ln 2)^2=\frac{-1}{8}(\ln 2)^2+C \
& \Rightarrow C=0 \
& \therefore y(\ln (\sec x))^3=\frac{-1}{2}(\ln (\sec x))^2+0 \
& y=\frac{-1}{2 \ln (\sec x)} \
& y=\frac{1}{2 \ln (\cos x)}
\end{aligned}$$

$$\begin{aligned}
& \therefore y\left(\frac{\pi}{6}\right)=\frac{1}{2 \ln \left(\cos \frac{\pi}{6}\right)} \
& =\frac{1}{2 \ln \left(\frac{\sqrt{3}}{2}\right)} \
& =\frac{1}{2\left(\frac{1}{2} \ln 3-\ln 2\right)} \
& =\frac{1}{\ln 3-\ln 4}
\end{aligned}$$

Solutions · 1