If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :
Solution
<p>$$\begin{aligned}
& \frac{d x}{d y}+\frac{x}{1+y^2}=\frac{2 e^{\tan ^{-1} y}}{1+y^2} \\
& \text { I.F. }=e^{\tan ^{-1} y} \\
& x^{\tan ^{-1} y}=\int \frac{2\left(e^{\tan ^{-1} y}\right)^2 d y}{1+y^2} \\
& \text { Put } \tan ^{-1} y=t, \frac{d y}{1+y^2}=d t \\
& x e^{\tan ^{-1} y}=\int 2 e^{2 t} d t \\
& x e^{\tan ^{-1} y}=e^{2 \tan ^{-1} y}+c \\
& x=e^{\tan ^{-1} y}+c e^{-\tan ^{-1} y} \\
& \because y=0, x=1 \\
& 1=1+c \Rightarrow c=0 \\
& y=\frac{1}{\sqrt{3}}, x=e^{\pi / 6}
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Differential Equations · Topic: Order and Degree
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