# I 2nd order non-linear ODE

1. Feb 1, 2017

### joshmccraney

Can someone check my work here? Both $f=f(x)$ and $y=y(x)$.
$$f'y'+\frac{fy''}{1+y'^2}=0\implies\\ \frac{y''}{y'(1+y'^2)}=-\frac{f'}{f}\\ \frac{y''}{y'(1+y'^2)}=-\ln(f)$$
Now let $v=y'$, which implies
$$\int\frac{1}{v(1+v^2)}\,dv=-\int\ln(f)\,dx\implies\\ \ln(v) - \frac{1}{2}\ln(1 + v^2)+C=-\int\ln(f)\,dx\implies\\ \frac{y'}{\sqrt{1 + y'^2}}=k\exp\left[-\int\ln(f)\,dx\right]$$
Am I correct to this point? Also, how would you proceed?

2. Feb 1, 2017

### the_wolfman

This last line is not correct. The right side should be $-\frac{d}{dx} \ln f\left(x\right)$

3. Feb 2, 2017

Thanks!!!