Internal problem ID [8361]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 24.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
\[ \boxed {y^{\prime }+a y^{2}=b \,x^{\nu }} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 214
dsolve(diff(y(x),x) + a*y(x)^2 - b*x^nu=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\operatorname {BesselJ}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-b a}\, x^{\frac {\nu }{2}+1} c_{1} -\operatorname {BesselY}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-b a}\, x^{\frac {\nu }{2}+1}+c_{1} \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{x a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-b a}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 770
DSolve[y'[x] + a*y[x]^2 - b*x^nu == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} (-1)^{\frac {1}{\nu +2}} x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} (-1)^{\frac {1}{\nu +2}} x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (1+\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+(-1)^{\frac {1}{\nu +2}} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+c_1 \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{2 a x \left ((-1)^{\frac {1}{\nu +2}} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+c_1 \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )} y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{\nu /2} \left (\operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )}{\operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}+\frac {1}{x}}{2 a} y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{\nu /2} \left (\operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )}{\operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}+\frac {1}{x}}{2 a} \end{align*}