# First order non-linear ordinary differential equation

The following ODE is given:

$a\pm\sqrt{b+c*(x(t)+d))}=e*x'(t)+f*x(t)$

from Matlab I'm able to get a solution for the differential equation (actually two solutions, one for the + and one for the - sign) such as "solve(eqn)", but I'm not able to solve the eqn inside of the solve.

Looking at the solve.m file I found out that Matlab is redirecting to MuPAD, but I don't know how to access this tool or if it is actually worth it.

Can you solve the equation for x(t) or give me any clue for which methods/tools I could use for doing it?

Let $$u = b + x(x+d)$$ then we find $$\frac{e}{c}u' + \frac{f}{c}u - \lambda = \pm\sqrt{u}$$ where $$\lambda = -\left[f\left(\frac{b+dc}{c}\right)+a\right]$$ then $u = v^2$ we find $$\frac{2e}{c}vv' + \frac{f}{c}v^2 - \lambda = \pm v$$ this is a separable equation namely $$\frac{2v}{v^2 \mp \frac{c}{f}v - \frac{c}{f}\lambda}dv = -\frac{f}{c}dt$$