Internal problem ID [4168]
Book: Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section: Abel ODE’s with constant invariant
Problem number: problem 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _Abel]
Solve \begin {gather*} \boxed {a x y^{3}+b y^{2}+y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 103
dsolve(a*x*y(x)^3+b*y(x)^2+diff(y(x),x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\RootOf \left (2 \sqrt {b^{2}+4 a}\, b \arctanh \left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}+4 a}}\right )-\ln \left (x^{2} \left ({\mathrm e}^{2 \textit {\_Z}} a +b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) b^{2}+2 c_{1} b^{2}+2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left ({\mathrm e}^{2 \textit {\_Z}} a +b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) a +8 c_{1} a +8 a \textit {\_Z} \right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.292 (sec). Leaf size: 103
DSolve[a*x*y[x]^3+b*y[x]^2+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {b^2 \left (\frac {2 \text {ArcTan}\left (\frac {-2 a x y(x)-b}{b \sqrt {-\frac {4 a}{b^2}-1}}\right )}{\sqrt {-\frac {4 a}{b^2}-1}}-\log \left (\frac {a (-x) y(x) (-a x y(x)-b)-a}{a^2 x^2 y(x)^2}\right )\right )}{2 a}=-\frac {b^2 \log (x)}{a}+c_1,y(x)\right ] \]