Internal problem ID [7618]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 37.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(diff(y(x),x) - y(x)^3 - a*exp(x)*y(x)^2=0,y(x), singsol=all)
\[ c_{1}+\frac {{\mathrm e}^{-\frac {\left (a \,{\mathrm e}^{x}+\frac {1}{y \relax (x )}\right )^{2}}{2}} {\mathrm e}^{-x}}{a}+\frac {\erf \left (\frac {\left (a \,{\mathrm e}^{x}+\frac {1}{y \relax (x )}\right ) \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\pi }}{2} = 0 \]
✓ Solution by Mathematica
Time used: 1.172 (sec). Leaf size: 78
DSolve[y'[x] - y[x]^3 - a*Exp[x]*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-i a e^x=\frac {2 e^{\frac {1}{2} \left (-i a e^x-\frac {i}{y(x)}\right )^2}}{\sqrt {2 \pi } \text {Erfi}\left (\frac {-i a e^x-\frac {i}{y(x)}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]