Internal problem ID [4169]
Book: Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section: Abel ODE’s with constant invariant
Problem number: problem 46.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1008
dsolve(diff(y(x),x)-x^a*y(x)^3+3*y(x)^2-x^(-a)*y(x)-x^(-2*a)+a*x^(-a-1) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {{\mathrm e}^{\frac {2 x \,x^{-a}}{a -1}}}{\sqrt {c_{1}-\frac {2 \,2^{-\frac {2 a}{-a +1}-\frac {2}{-a +1}} \left (\frac {1}{-a +1}\right )^{\frac {a +1}{a -1}} \left (-\frac {2^{-3+\frac {2 a}{-a +1}+\frac {2}{-a +1}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{-a +1}+\frac {1}{-a +1}-1+a} \left (\frac {1}{-a +1}\right )^{-\frac {a +1}{a -1}} \left (-\frac {4 x^{-a +1} a^{2}}{-a +1}+\frac {8 a \,x^{-a +1}}{-a +1}-\frac {4 x^{-a +1}}{-a +1}+2 a -2\right ) \left (-a +1\right ) \left (\frac {x^{-a +1}}{-a +1}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{-a +1}}{-a +1}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{-a +1}}{-a +1}\right )}{\left (a +1\right ) \left (-3+a \right )}+\frac {2^{-1+\frac {2 a}{-a +1}+\frac {2}{-a +1}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{-a +1}+\frac {1}{-a +1}-1+a} \left (\frac {1}{-a +1}\right )^{-\frac {a +1}{a -1}} \left (-a +1\right ) \left (\frac {x^{-a +1}}{-a +1}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{-a +1}}{-a +1}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}+1, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{-a +1}}{-a +1}\right )}{\left (a +1\right ) \left (-3+a \right )}\right )}{-a +1}}}+x^{-a} \\ y \relax (x ) = \frac {{\mathrm e}^{\frac {2 x \,x^{-a}}{a -1}}}{\sqrt {c_{1}-\frac {2 \,2^{-\frac {2 a}{-a +1}-\frac {2}{-a +1}} \left (\frac {1}{-a +1}\right )^{\frac {a +1}{a -1}} \left (-\frac {2^{-3+\frac {2 a}{-a +1}+\frac {2}{-a +1}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{-a +1}+\frac {1}{-a +1}-1+a} \left (\frac {1}{-a +1}\right )^{-\frac {a +1}{a -1}} \left (-\frac {4 x^{-a +1} a^{2}}{-a +1}+\frac {8 a \,x^{-a +1}}{-a +1}-\frac {4 x^{-a +1}}{-a +1}+2 a -2\right ) \left (-a +1\right ) \left (\frac {x^{-a +1}}{-a +1}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{-a +1}}{-a +1}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{-a +1}}{-a +1}\right )}{\left (a +1\right ) \left (-3+a \right )}+\frac {2^{-1+\frac {2 a}{-a +1}+\frac {2}{-a +1}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{-a +1}+\frac {1}{-a +1}-1+a} \left (\frac {1}{-a +1}\right )^{-\frac {a +1}{a -1}} \left (-a +1\right ) \left (\frac {x^{-a +1}}{-a +1}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{-a +1}}{-a +1}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}+1, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{-a +1}}{-a +1}\right )}{\left (a +1\right ) \left (-3+a \right )}\right )}{-a +1}}}+x^{-a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 16.62 (sec). Leaf size: 149
DSolve[y'[x]-x^a*y[x]^3+3*y[x]^2-x^(-a)*y[x]-x^(-2*a)+a*x^(-a-1) == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{-a}-\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2 x^{a+1} \text {ExpIntegralE}\left (\frac {2 a}{a-1},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a}+\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2 x^{a+1} \text {ExpIntegralE}\left (\frac {2 a}{a-1},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a} \\ \end{align*}