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The solution of the differential equation $y(1+\log x) \frac{d x}{d y}-x \log x=0$ is
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Verified Answer
The correct answer is:
$x \log x=y c$
Given differential equation is
$$
y(1+\log x) \frac{d x}{d y}-x \log x=0
$$
$$
\begin{array}{ll}
\Rightarrow & \int \frac{1+\log x}{x \log x} d x=\int \frac{d y}{y} \\
\Rightarrow & \int \frac{1}{x \log x} d x+\int \frac{1}{x} d x=\int \frac{1}{y} d y \\
\Rightarrow & \log (\log x)+\log x=\log y+\log c \\
\Rightarrow & x \log x=y c
\end{array}
$$
$$
y(1+\log x) \frac{d x}{d y}-x \log x=0
$$
$$
\begin{array}{ll}
\Rightarrow & \int \frac{1+\log x}{x \log x} d x=\int \frac{d y}{y} \\
\Rightarrow & \int \frac{1}{x \log x} d x+\int \frac{1}{x} d x=\int \frac{1}{y} d y \\
\Rightarrow & \log (\log x)+\log x=\log y+\log c \\
\Rightarrow & x \log x=y c
\end{array}
$$
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