Internal problem ID [4172]
Book: Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section: Abel ODE’s with constant invariant
Problem number: problem 169.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Abel]
Solve \begin {gather*} \boxed {\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 153
dsolve((a*x+b)^2*diff(y(x),x)+(a*x+b)*y(x)^3+c*y(x)^2 = 0,y(x), singsol=all)
\[ c_{1}+\left (x +\frac {b}{a}+\frac {c \sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \left (a^{2} x +a b +c y \relax (x )\right )}{2 \sqrt {a}\, y \relax (x ) \left (a x +b \right )}\right ) {\mathrm e}^{\frac {\left (a^{2} x +a b +c y \relax (x )\right )^{2}}{2 y \relax (x )^{2} \left (a x +b \right )^{2} a}}}{2 a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (a^{2} x +a x y \relax (x )+a b +b y \relax (x )+c y \relax (x )\right ) \left (a^{2} x -a x y \relax (x )+a b -b y \relax (x )+c y \relax (x )\right )}{2 y \relax (x )^{2} \left (a x +b \right )^{2} a}} = 0 \]
✓ Solution by Mathematica
Time used: 1.383 (sec). Leaf size: 149
DSolve[(a*x+b)^2*y'[x]+(a*x+b)*y[x]^3+c*y[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {c}{\sqrt {-a (a x+b)^2}}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}\right )^2\right )}{\sqrt {2 \pi } \text {Erfi}\left (\frac {-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]