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The solution of the differential equation $\frac{d y}{d x}=\frac{y}{x}+\frac{\phi(y / x)}{\phi^{\prime}(y / x)}$ is
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Verified Answer
The correct answer is:
$\phi\left(\frac{y}{x}\right)=k x$
Given, $\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}$
Put $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
$\begin{aligned}
& \therefore \quad v+x \frac{d v}{d x}=v+\frac{\phi(v)}{\phi^{\prime}(v)} \\
& \Rightarrow \quad \frac{\phi^{\prime}(v)}{\phi(v)} d v=\frac{d x}{x} \\
&
\end{aligned}$
On integrating, we get
$\begin{gathered}
\log \phi(v)=\log x+\log k \\
\Rightarrow \quad \phi(v)=k x \Rightarrow \phi\left(\frac{y}{x}\right)=k x
\end{gathered}$
Put $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
$\begin{aligned}
& \therefore \quad v+x \frac{d v}{d x}=v+\frac{\phi(v)}{\phi^{\prime}(v)} \\
& \Rightarrow \quad \frac{\phi^{\prime}(v)}{\phi(v)} d v=\frac{d x}{x} \\
&
\end{aligned}$
On integrating, we get
$\begin{gathered}
\log \phi(v)=\log x+\log k \\
\Rightarrow \quad \phi(v)=k x \Rightarrow \phi\left(\frac{y}{x}\right)=k x
\end{gathered}$
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