# First order separable differential equations (d^2y)/(dx^2)+2(dy)/(xdx) = 0 Using change of variable equals to Z=(dy)/(dx) what is the 1st order separable differential equation for Z as function of x? and solve for y(x).

First order separable differential equations
$\frac{{d}^{2}y}{d{x}^{2}}+2\frac{dy}{xdx}=0$
Using change of variable equals to $Z=\frac{dy}{dx}$ what is the 1st order separable differential equation for Z as function of x?
and solve for y(x).
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Jonah Cooke
Let $Z=\frac{dy}{dx}$. Then $\frac{{d}^{2}y}{d{x}^{2}}+2\frac{dy}{xdx}=0$ can be written as
$\frac{dZ}{dx}+2\frac{Z}{x}=0.$
Then $\frac{dZ}{Z}=-2\frac{dx}{x}$
$\int \frac{dZ}{Z}=-\int 2\frac{dx}{x}$
So it's easy to get $Z=\frac{C}{{x}^{2}}$, where C is a constant. Then for $Z=\frac{dy}{dx}=\frac{C}{{x}^{2}}$. We get the solution is $y=-\frac{C}{x}+{C}_{0}$, where ${C}_{0}$ is a constant.