Internal problem ID [8475]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 139.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )=-a \,x^{k}+b \left (b -1\right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 296
dsolve(x^2*(diff(y(x),x)+y(x)^2) + a*x^k - b*(b-1)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\sqrt {a}\, x^{\frac {k}{2}} c_{1} \operatorname {BesselY}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )}{x \left (\operatorname {BesselY}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}+\frac {\left (\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right ) c_{1} +c_{1} \right ) \operatorname {BesselY}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )-2 \operatorname {BesselJ}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \sqrt {a}\, x^{\frac {k}{2}}+\left (\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )+1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )}{2 x \left (\operatorname {BesselY}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.496 (sec). Leaf size: 627
DSolve[x^2*(y'[x]+y[x]^2) + a*x^k - b*(b-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} x^k \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (-\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\sqrt {a} x^k \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+\sqrt {x^k} \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\sqrt {a} c_1 x^k \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+\sqrt {a} c_1 x^k \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_1 \sqrt {x^k} \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}{2 x \sqrt {x^k} \left (\operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_1 \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )} y(x)\to \frac {\frac {\sqrt {a} \sqrt {x^k} \left (\operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )}{\operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}+1}{2 x} y(x)\to \frac {\frac {\sqrt {a} \sqrt {x^k} \left (\operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )}{\operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}+1}{2 x} \end{align*}