Let y=y(x) be the solution of x(_e( x))^2,dy+( x-3y x_e( x)),dx=0, x(0,/2).

Question

Let $y=y(x)$ be the solution of $\cos x(\log_e(\cos x))^2\,dy+(\sin x-3y\sin x\log_e(\cos x))\,dx=0$, $x\in(0,\pi/2)$. If $y\!\left(\dfrac{\pi}{4}\right)-\dfrac{1}{\log_e2}y\!\left(\dfrac{\pi}{6}\right)=2\lambda$, then $\lambda$ is:

Accepted Answer

$$\frac{dy}{dx}-\frac{3\tan x}{\ln(\cos x)}y = \frac{-\tan x}{(\ln(\cos x))^2}.$$

I.F. $=(\ln(\sec x))^3$.
$$y(\ln(\sec x))^3 = -\frac12(\ln(\sec x))^2+C.$$

The given condition fixes $C=0$, giving $y=\dfrac{-1}{2\ln(\sec x)}=\dfrac{1}{2\ln(\cos x)}$.

$$y\!\left(\frac{\pi}{6}\right) = \frac{1}{2}\ln\!\left(\frac{\sqrt3}{2}\right) = \frac{1}{\ln3-\ln4}.$$

Solutions · 1