Internal problem ID [8436]
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]
\[ \boxed {y^{\prime } x +a y^{2}-b y=c \,x^{\beta }} \]
✓ 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)
\[ y \left (x \right ) = -\frac {\sqrt {-a c}\, x^{\frac {\beta }{2}} c_{1} \operatorname {BesselY}\left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}-\frac {\operatorname {BesselJ}\left (\frac {b +\beta }{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) \sqrt {-a c}\, x^{\frac {\beta }{2}}-\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} b -b \operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.322 (sec). Leaf size: 428
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 {\sqrt {-a} \sqrt {c} x^{\beta /2} \left (-2 \operatorname {BesselJ}\left (\frac {b}{\beta }-1,\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 \left (\operatorname {BesselJ}\left (1-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )-\operatorname {BesselJ}\left (-\frac {b+\beta }{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )\right )-b c_1 \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )} y(x)\to \frac {-\sqrt {-a} \sqrt {c} x^{\beta /2} \operatorname {BesselJ}\left (1-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+\sqrt {-a} \sqrt {c} x^{\beta /2} \operatorname {BesselJ}\left (-\frac {b+\beta }{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+b \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )} \end{align*}