Practicing Success
If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x=2 x \sec x$ and $y(0)=0$, then |
$y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{8 \sqrt{2}}, y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^2}{3 \sqrt{3}}$ $y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{4 \sqrt{2}}, y^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi^2}{18}$ $y\left(\frac{\pi}{3}\right)=\frac{\pi^2}{9}, y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{\pi^2}{3 \sqrt{3}}$ $y\left(\frac{\pi}{3}\right)=\frac{\pi^2}{4 \sqrt{2}}, y^{\prime}\left(\frac{\pi}{3}\right)=\frac{\pi^2}{18}$ |
$y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{8 \sqrt{2}}, y^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^2}{3 \sqrt{3}}$ |
We have, $\frac{d y}{d x}-(\tan x) y=2 x \sec x$ ......(i) It is a linear differential equation with Integrating factor = $e^{-\int \tan x d x}=e^{\log \cos x}=\cos x$ Multiplying (i) by integrating factor = $\cos x$ and integrating, we get $\Rightarrow y \cos x=x^2+C$ ........(ii) Putting $x=0$ and $y=0$, we get $0=0+C \Rightarrow C=0$ Putting $C=0$ in (ii), we get $y =x^2 \sec x$ ........(iii) $\Rightarrow y' =2 x \sec x+x^2 \sec x \tan x$ ........(iv) Putting $x=\frac{\pi}{4}$ in (iii) and (iv), we get $y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{8 \sqrt{2}}, y'\left(\frac{\pi}{4}\right)=\frac{\pi}{\sqrt{2}}+\frac{\pi^2}{8 \sqrt{2}}$ Putting $x=\frac{\pi}{3}$ in (iii) and (iv), we get $y\left(\frac{\pi}{3}\right)=\frac{2 \pi^2}{9}, y'\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^2}{3 \sqrt{3}}$ |