Internal problem ID [4680]
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]
\[ \boxed {\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 126
dsolve((a*x+b)^2*diff(y(x),x)+(a*x+b)*y(x)^3+c*y(x)^2 = 0,y(x), singsol=all)
\[ \frac {\left (\sqrt {a}\, b +a^{\frac {3}{2}} x \right ) {\mathrm e}^{-\frac {\left (\left (a x +b +c \right ) y \left (x \right )+a \left (a x +b \right )\right ) \left (\left (-a x -b +c \right ) y \left (x \right )+a \left (a x +b \right )\right )}{2 y \left (x \right )^{2} \left (a x +b \right )^{2} a}}+\frac {c \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2 a}} \operatorname {erf}\left (\frac {\left (c y \left (x \right )+a \left (a x +b \right )\right ) \sqrt {2}}{2 \sqrt {a}\, y \left (x \right ) \left (a x +b \right )}\right )}{2}+c_{1} a^{\frac {3}{2}}}{a^{\frac {3}{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 1.43 (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]
\[ \text {Solve}\left [-\frac {c}{\sqrt {-a (a x+b)^2}}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}\right )^2\right )}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {c}{\sqrt {-a (a x+b)^2}}-\frac {\left (-a (a x+b)^2\right )^{3/2}}{a y(x) (a x+b)^3}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]