Internal problem ID [7724]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 144.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }+a y^{2}\right )+b \,x^{\alpha }+c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 244
dsolve(x^2*(diff(y(x),x)+a*y(x)^2) + b*x^alpha + c=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-\sqrt {-4 a c +1}\, c_{1}-c_{1}\right ) \BesselY \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )+2 \sqrt {a b}\, x^{\frac {\alpha }{2}} \BesselY \left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1}+\left (-\sqrt {-4 a c +1}-1\right ) \BesselJ \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )+2 \BesselJ \left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) \sqrt {a b}\, x^{\frac {\alpha }{2}}}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.589 (sec). Leaf size: 1777
DSolve[x^2*(y'[x]+a*y[x]^2) + b*x^\[Alpha] + c==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+1} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}}-a^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+1} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}}+a^{\frac {i \sqrt {4 a c-1}}{\alpha }} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }}+i a^{\frac {i \sqrt {4 a c-1}}{\alpha }} \sqrt {4 a c-1} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }}-a^{\frac {i \sqrt {4 a c-1}}{\alpha }} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }+\frac {1}{2}} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}} \sqrt {(1-4 a c) \alpha ^2} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {i \sqrt {4 a c-1}}{\alpha }}+a^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}} \left (x^{\alpha }\right )^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+\frac {1}{2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }} \left (-i \sqrt {4 a c-1} \alpha +\alpha +\sqrt {(1-4 a c) \alpha ^2}\right ) J_{-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \text {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}+a^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} J_{-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \text {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+\frac {1}{2}}-a^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+1} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }+1} J_{1-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \text {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) b^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}+\frac {1}{2}}}{2 a x \sqrt {x^{\alpha }} \alpha \left (a^{\frac {i \sqrt {4 a c-1}}{\alpha }} b^{\frac {i \sqrt {4 a c-1}}{\alpha }} \alpha ^{\frac {2 \sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}} J_{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \text {Gamma}\left (\frac {\alpha +\sqrt {1-4 a c}}{\alpha }\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1}}{\alpha }}+a^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}} b^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}} \alpha ^{\frac {2 i \sqrt {4 a c-1}}{\alpha }} J_{-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \text {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) \left (x^{\alpha }\right )^{\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\right )} \\ y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \sqrt {x^{\alpha }} \left (J_{-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{1-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right )}{J_{-\frac {\sqrt {(1-4 a c) \alpha ^2}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )}+\frac {\sqrt {\alpha ^2 (1-4 a c)}}{\alpha }-i \sqrt {4 a c-1}+1}{2 a x} \\ \end{align*}