If y=y(x) is the solution curve of the differential equation [(d y)/(d x)+y x=x x, 0 x ()/…

Question

If $y=y(x)$ is the solution curve of the differential equation

$$\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to

Accepted Answer

$\frac{d y}{d x}+y \tan x=x \sec x$

$\therefore $ I.F $=e^{\int \tan x d x}=\sec x$

$\Rightarrow y \sec x=\int x \sec ^{2} x d x$

$\Rightarrow y \sec x=x \tan x-\ln |\sec x|+c$

Given, $y(0)=1$

$\Rightarrow 1=c$

$$ \therefore $$ $ y \sec x=x \tan x-\ln |\sec x|+1$

$\Rightarrow y=x \sin x-\cos x \ln |\sec x|+cosx$

Now, at $x=\pi / 6$, we have

$$
\begin{aligned}
& y=\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _e \frac{2}{\sqrt{3}}+\frac{\sqrt{3}}{2} \\
& =\frac{\pi}{12}-\frac{\sqrt{3}}{2}\left[\log _e \frac{2}{\sqrt{3}}-\log _e e\right]\\
&=\frac{\pi}{12}-\frac{\sqrt{3}}{2} \log _e\left(\frac{2}{e \sqrt{3}}\right)
\end{aligned}
$$

Solutions · 1