Problem in the Differential equation

Question

I have the following Differential equation which I need to solve:

$$\frac{\mathrm{d^2} y}{\mathrm{d} x^2} = \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2$$

I know how to solve a second order linear differential equations but this is something strange equation which I have seen before while practicing the Differential equation Chapter. Please tell me how to go about solving this Differential equation problem. Any intial hint would do for me.

Thanks

Answer 1

Put $\frac{dy}{dx}$ = $v$, after differentiating w.r.to $x$ you get $\frac{dv}{dx}$ = $v^2$

then $\frac{-1}{v}$ = $x+C$

or, $v$ = $\frac{-1}{x+C}$

so $\frac{dy}{dx}$ = $\frac{-1}{x+C}$

$y$ = $-ln(x+C)$ + K