Internal problem ID [7627]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 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: 0.001 (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}{1-a}-\frac {2}{1-a}} \left (\frac {1}{1-a}\right )^{\frac {a +1}{a -1}} \left (-\frac {2^{-3+\frac {2 a}{1-a}+\frac {2}{1-a}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{1-a}+\frac {1}{1-a}-1+a} \left (\frac {1}{1-a}\right )^{-\frac {a +1}{a -1}} \left (-\frac {4 x^{1-a} a^{2}}{1-a}+\frac {8 a \,x^{1-a}}{1-a}-\frac {4 x^{1-a}}{1-a}+2 a -2\right ) \left (1-a \right ) \left (\frac {x^{1-a}}{1-a}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{1-a}}{1-a}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{1-a}}{1-a}\right )}{\left (a +1\right ) \left (-3+a \right )}+\frac {2^{-1+\frac {2 a}{1-a}+\frac {2}{1-a}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{1-a}+\frac {1}{1-a}-1+a} \left (\frac {1}{1-a}\right )^{-\frac {a +1}{a -1}} \left (1-a \right ) \left (\frac {x^{1-a}}{1-a}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{1-a}}{1-a}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}+1, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{1-a}}{1-a}\right )}{\left (a +1\right ) \left (-3+a \right )}\right )}{1-a}}}+x^{-a} \\ y \relax (x ) = \frac {{\mathrm e}^{\frac {2 x \,x^{-a}}{a -1}}}{\sqrt {c_{1}-\frac {2 \,2^{-\frac {2 a}{1-a}-\frac {2}{1-a}} \left (\frac {1}{1-a}\right )^{\frac {a +1}{a -1}} \left (-\frac {2^{-3+\frac {2 a}{1-a}+\frac {2}{1-a}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{1-a}+\frac {1}{1-a}-1+a} \left (\frac {1}{1-a}\right )^{-\frac {a +1}{a -1}} \left (-\frac {4 x^{1-a} a^{2}}{1-a}+\frac {8 a \,x^{1-a}}{1-a}-\frac {4 x^{1-a}}{1-a}+2 a -2\right ) \left (1-a \right ) \left (\frac {x^{1-a}}{1-a}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{1-a}}{1-a}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{1-a}}{1-a}\right )}{\left (a +1\right ) \left (-3+a \right )}+\frac {2^{-1+\frac {2 a}{1-a}+\frac {2}{1-a}+\frac {2}{a -1}} \left (a -1\right ) x^{-\frac {a^{2}}{1-a}+\frac {1}{1-a}-1+a} \left (\frac {1}{1-a}\right )^{-\frac {a +1}{a -1}} \left (1-a \right ) \left (\frac {x^{1-a}}{1-a}\right )^{\frac {1}{a -1}} {\mathrm e}^{-\frac {2 x^{1-a}}{1-a}} \WhittakerM \left (-\frac {a +1}{a -1}+\frac {1}{a -1}+1, -\frac {1}{a -1}+\frac {1}{2}, \frac {4 x^{1-a}}{1-a}\right )}{\left (a +1\right ) \left (-3+a \right )}\right )}{1-a}}}+x^{-a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.076 (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} E_{\frac {2 a}{a-1}}\left (-\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} E_{\frac {2 a}{a-1}}\left (-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a} \\ \end{align*}