Internal problem ID [7605]
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]]
Solve \begin {gather*} \boxed {y^{\prime }+a y^{2}-b \,x^{\nu }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 216
dsolve(diff(y(x),x) + a*y(x)^2 - b*x^nu=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\BesselJ \left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-a b}\, x^{\frac {\nu }{2}+1} c_{1}+\BesselY \left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-a b}\, x^{\frac {\nu }{2}+1}-c_{1} \BesselJ \left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )-\BesselY \left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{x a \left (c_{1} \BesselJ \left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\BesselY \left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.251 (sec). Leaf size: 271
DSolve[y'[x] + a*y[x]^2 - b*x^nu == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(\nu +2) \left (\frac {\sqrt {-a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )^{\frac {2}{\nu +2}} \, _0\tilde {F}_1\left (;\frac {1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )+c_1 (\nu +2) \, _0\tilde {F}_1\left (;-\frac {1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )+c_1 \, _0\tilde {F}_1\left (;\frac {\nu +1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )}{a x \left (\left (\frac {\sqrt {-a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )^{\frac {2}{\nu +2}} \, _0\tilde {F}_1\left (;1+\frac {1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )+c_1 \, _0\tilde {F}_1\left (;\frac {\nu +1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )\right )} \\ y(x)\to \frac {1-\frac {\, _0F_1\left (;-\frac {1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )}{\, _0F_1\left (;\frac {\nu +1}{\nu +2};\frac {a b x^{\nu +2}}{(\nu +2)^2}\right )}}{a x} \\ \end{align*}