For $x0$ and $y0$, solve $\frac{\mathrm dy}{\mathrm dx}=\frac yx+\sqrt{\frac yx}$ subject to

Question

For $x>0$ and $y>0$, solve $\frac{\mathrm dy}{\mathrm dx}=\frac yx+\sqrt{\frac yx}$ subject to $y(1)=1$. State the interval of the particular solution and find $y(e^2)$.

Accepted Answer

$y=x(1+\tfrac12\ln x)^2\ (x>e^{-2});\quad y(e^2)=4e^2$

Homogeneous differential equation with a square root

Homogeneous differential equation with a square root

Solution:

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