Internal problem ID [8367]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 30.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+x^{-a -1} y^{2}=x^{a}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 81
dsolve(diff(y(x),x) + x^(-a-1)*y(x)^2 - x^a=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {c_{1} x^{a +1} \operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right )}{\sqrt {x}\, \left (\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right )\right )}+\frac {\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right ) x^{a +1}}{\sqrt {x}\, \left (\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.363 (sec). Leaf size: 265
DSolve[y'[x] + x^(-a-1)*y[x]^2 - x^a==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a-1,2 \sqrt {x}\right )+\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (1-a,2 \sqrt {x}\right )-a \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-(-1)^a a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \left (\operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )\right )} y(x)\to \frac {x^a \left (\sqrt {x} \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-a \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+\sqrt {x} \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \operatorname {BesselI}\left (a,2 \sqrt {x}\right )} \end{align*}