Internal problem ID [7680]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 99.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }+a y^{2}-b y-c \,x^{\beta }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 237
dsolve(x*diff(y(x),x) + a*y(x)^2 - b*y(x) - c*x^beta=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {x^{\frac {\beta }{2}} \sqrt {-a c}\, c_{1} \BesselY \left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )}{a \left (\BesselY \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1}+\BesselJ \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}-\frac {-\BesselY \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} b +\BesselJ \left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) \sqrt {-a c}\, x^{\frac {\beta }{2}}-b \BesselJ \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )}{a \left (\BesselY \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1}+\BesselJ \left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.571 (sec). Leaf size: 210
DSolve[x*y'[x] + a*y[x]^2 - b*y[x] - c*x^\[Beta]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\beta ^2 \left (\frac {\sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )^{\frac {2 b}{\beta }} \, _0\tilde {F}_1\left (;\frac {b}{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )+a c c_1 x^{\beta } \, _0\tilde {F}_1\left (;2-\frac {b}{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )}{a \beta \left (\left (\frac {\sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )^{\frac {2 b}{\beta }} \, _0\tilde {F}_1\left (;\frac {b+\beta }{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )+c_1 \, _0\tilde {F}_1\left (;1-\frac {b}{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )\right )} \\ y(x)\to \frac {c x^{\beta } \, _0\tilde {F}_1\left (;2-\frac {b}{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )}{\beta \, _0\tilde {F}_1\left (;1-\frac {b}{\beta };\frac {a c x^{\beta }}{\beta ^2}\right )} \\ \end{align*}