33.1.2 problem problem 41

Internal problem ID [6020]
Book : Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section : Abel ODE with constant invariant
Problem number : problem 41
Date solved : Monday, January 27, 2025 at 01:32:46 PM
CAS classification : [[_homogeneous, `class G`], _Abel]

\begin{align*} a x y^{3}+b y^{2}+y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 103

dsolve(a*x*y(x)^3+b*y(x)^2+diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {b^{2}+4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}+4 a}}\right )-\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) b^{2}+2 c_1 \,b^{2}+2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) a +8 c_1 a +8 \textit {\_Z} a \right )}}{x} \]

✓ Solution by Mathematica

Time used: 0.189 (sec). Leaf size: 103

DSolve[a*x*y[x]^3+b*y[x]^2+D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\frac {b^2 \left (\frac {2 \arctan \left (\frac {-2 a x y(x)-b}{b \sqrt {-\frac {4 a}{b^2}-1}}\right )}{\sqrt {-\frac {4 a}{b^2}-1}}-\log \left (\frac {a (-x) y(x) (-a x y(x)-b)-a}{a^2 x^2 y(x)^2}\right )\right )}{2 a}=-\frac {b^2 \log (x)}{a}+c_1,y(x)\right ] \]