Internal problem ID [8437]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 100.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Riccati, _special]]
\[ \boxed {y^{\prime } x +x y^{2}=-a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 59
dsolve(x*diff(y(x),x) + x*y(x)^2 + a=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.224 (sec). Leaf size: 289
DSolve[x*y'[x] + x*y[x]^2 + a==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (0,2 \sqrt {a} \sqrt {x}\right )+2 \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (2,2 \sqrt {a} \sqrt {x}\right )-i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )-i c_1 \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )+i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{4 x \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 i c_1 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )} y(x)\to \frac {\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )+\operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )-\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{2 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )} \end{align*}