Nonlinear second order differential equation

[JulieK]

What is the solution of the follwoing differential equation

$\frac{\partial^{2}y}{\partial x^{2}}-ay^{-1}\frac{dy}{dx}=0$

where $a$ is a constant.

 

[pasmith]

What is the solution of the follwoing differential equation

$\frac{\partial^{2}y}{\partial x^{2}}-ay^{-1}\frac{dy}{dx}=0$

where $a$ is a constant.

Use the identity
$$\frac{\mathrm{d}^2y}{\mathrm{d} x^2} = \frac{\mathrm{d}y'}{\mathrm{d}x} = \frac{\mathrm{d}y'}{\mathrm{d}y}\frac{\mathrm{d}y}{\mathrm{d}x} = y'\frac{\mathrm{d}y'}{\mathrm{d}y}$$
to find $y' = \frac{\mathrm{d}y}{\mathrm{d}x}$ as a function of $y$. Then use separation of variables.

 

[JulieK]

Thank you