solve the diffrential equation:

Question

Solve the DE

y^''= e^x (y^')²

Answer 1

Let y' = v, then y'' = v' and we get ODE for functionv: v' = e^xv²;

We separate variables and integrate: ∫v^-2dv = ∫e^xdx; -1/v = e^x + C₁; v = - 1/(e^x + C₁)

y' = - 1/(e^x + C₁); ∫dy = ∫-1/(e^x + C₁)dx; y = 1/C₁(lnIe^x + C₁I - x) + C₂