# What is a particular solution to the differential equation dy/dx=4√ylnx/x with y(e)=1?

What is a particular solution to the differential equation $\frac{dy}{dx}=\frac{4\sqrt{y}\mathrm{ln}x}{x}$ with y(e)=1?
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Clarence Mills
We have:

$\frac{dy}{dx}=\frac{4\sqrt{y}\mathrm{ln}x}{x}$

Which is a first order linear separable Differential Equation, so we can rearrange to get:

$\frac{1}{\sqrt{y}}\frac{dy}{dx}=\frac{4\mathrm{ln}x}{x}$

and separate the variables to get:

And then we can integrate to get:

$\frac{{y}^{\frac{1}{2}}}{\frac{1}{2}}=\left(4\right)\left(\frac{{\mathrm{ln}}^{2}x}{2}\right)+C$
$\therefore 2\sqrt{y}=2{\mathrm{ln}}^{2}x+C$

Using y(e)=1 we get:

$\therefore 2=2{\mathrm{ln}}^{2}e+C$
$\therefore C=0$

Hence the particular solution is:

$\therefore \sqrt{y}={\mathrm{ln}}^{2}x$

Validation:
1. $x=e⇒y={\mathrm{ln}}^{4}e=1$ QED
2. $\frac{dy}{dx}=\frac{4{\mathrm{ln}}^{3}x}{x}=\frac{4\mathrm{ln}x}{x}\cdot {\mathrm{ln}}^{2}x=\frac{4\sqrt{y}\mathrm{ln}x}{x}$ QED