Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y>0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :
Solution
<p>$$\begin{aligned}
\quad y d y & =(x d y-y d x) \sin \left(\frac{x}{y}\right) \\
\frac{d y}{y} & =\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\
\frac{d y}{y} & =\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\
\Rightarrow \quad \ell n y & =\cos \frac{x}{y}+C
\end{aligned}$$</p>
<p>$$\begin{aligned}
& x(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+C \Rightarrow C=0 \\
& \text { थny }=\cos \frac{x}{y} \\
& \text { but } y=2 \Rightarrow \cos \frac{x}{2}=\ln 2 \\
& \qquad \begin{aligned}
\cos x & =2 \cos ^2 \frac{x}{2}-1 \\
& =2(\ln 2)^2-1
\end{aligned}
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Differential Equations · Topic: Order and Degree
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