Internal problem ID [7688]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 107.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 176
dsolve(x*diff(y(x),x) + a*x^alpha*y(x)^2 + b*y(x) - c*x^beta=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\BesselY \left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1}+\BesselJ \left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) x^{\frac {\alpha }{2}+\frac {\beta }{2}} \sqrt {-a c}\, x^{1-\alpha }}{\left (\BesselY \left (-\frac {\alpha -b}{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1}+\BesselJ \left (-\frac {\alpha -b}{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 0.897 (sec). Leaf size: 518
DSolve[x*y'[x] + a*x^\[Alpha]*y[x]^2 + b*y[x] - c*x^\[Beta]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {c} (\alpha +\beta ) x^{\beta } \left ((-1)^{\frac {\alpha }{\alpha +\beta }} (\alpha +\beta )^{\frac {2 \alpha }{\alpha +\beta }} \left ((\alpha +\beta )^2\right )^{\frac {b}{\alpha +\beta }} \operatorname {Gamma}\left (\frac {\alpha -b}{\alpha +\beta }+1\right ) I_{-\frac {b+\beta }{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {(\alpha +\beta )^2}}\right )+c_1 \left ((\alpha +\beta )^2\right )^{\frac {\alpha }{\alpha +\beta }} (-1)^{\frac {b}{\alpha +\beta }} (\alpha +\beta )^{\frac {2 b}{\alpha +\beta }} \operatorname {Gamma}\left (\frac {b+\beta }{\alpha +\beta }\right ) I_{\frac {b+\beta }{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {(\alpha +\beta )^2}}\right )\right )}{\sqrt {a} \sqrt {(\alpha +\beta )^2} \sqrt {x^{\alpha +\beta }} \left ((-1)^{\frac {\alpha }{\alpha +\beta }} (\alpha +\beta )^{\frac {2 \alpha }{\alpha +\beta }} \left ((\alpha +\beta )^2\right )^{\frac {b}{\alpha +\beta }} \operatorname {Gamma}\left (\frac {\alpha -b}{\alpha +\beta }+1\right ) I_{\frac {\alpha -b}{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {(\alpha +\beta )^2}}\right )+c_1 \left ((\alpha +\beta )^2\right )^{\frac {\alpha }{\alpha +\beta }} (-1)^{\frac {b}{\alpha +\beta }} (\alpha +\beta )^{\frac {2 b}{\alpha +\beta }} \operatorname {Gamma}\left (\frac {b+\beta }{\alpha +\beta }\right ) I_{\frac {b-\alpha }{\alpha +\beta }}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{\alpha +\beta }}}{\sqrt {(\alpha +\beta )^2}}\right )\right )} \\ y(x)\to \frac {x^{-\alpha } \left (\frac {a c x^{\alpha +\beta } \, _0\tilde {F}_1\left (;\frac {b+\beta }{\alpha +\beta }+1;\frac {a c x^{\alpha +\beta }}{(\alpha +\beta )^2}\right )+(\alpha +\beta )^2 \, _0\tilde {F}_1\left (;\frac {b-\alpha }{\alpha +\beta };\frac {a c x^{\alpha +\beta }}{(\alpha +\beta )^2}\right )}{(\alpha +\beta ) \, _0\tilde {F}_1\left (;\frac {b+\beta }{\alpha +\beta };\frac {a c x^{\alpha +\beta }}{(\alpha +\beta )^2}\right )}+\alpha -b\right )}{2 a} \\ \end{align*}