# How to you find the general solution of ylnx−xy'=0?

How to you find the general solution of ylnx−xy'=0?
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allelog5
This is a linear homogeneous differential equation. Also is separable. So grouping variables,
$\frac{y\prime }{y}=\frac{{\mathrm{log}}_{e}x}{x}$ After integration
${\mathrm{log}}_{e}y=\frac{1}{2}{\left({\mathrm{log}}_{e}x\right)}^{2}+C$ so
$y={C}_{1}{e}^{\frac{1}{2}{\left({\mathrm{log}}_{e}x\right)}^{2}}$