Internal problem ID [7719]
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]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 300
dsolve(x^2*(diff(y(x),x)+y(x)^2) + a*x^k - b*(b-1)=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {x^{\frac {k}{2}} \sqrt {a}\, c_{1} \BesselY \left (\frac {\sqrt {\left (2 b -1\right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )}{x \left (\BesselY \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}-\frac {\left (-\mathrm {csgn}\left (2 b -1\right ) \left (2 b -1\right ) c_{1}-c_{1}\right ) \BesselY \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )+2 \BesselJ \left (\frac {\sqrt {\left (2 b -1\right )^{2}}+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \sqrt {a}\, x^{\frac {k}{2}}+\left (-\mathrm {csgn}\left (2 b -1\right ) \left (2 b -1\right )-1\right ) \BesselJ \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )}{2 x \left (\BesselY \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {\left (2 b -1\right )^{2}}}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.712 (sec). Leaf size: 470
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 {\text {Gamma}\left (\frac {2 b+k-1}{k}\right ) \left (b \sqrt {x^k} J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\sqrt {a} x^k J_{\frac {2 b+k-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )-\sqrt {a} c_1 x^k \text {Gamma}\left (\frac {-2 b+k+1}{k}\right ) J_{\frac {-2 b+k+1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-(b-1) c_1 \sqrt {x^k} \text {Gamma}\left (\frac {-2 b+k+1}{k}\right ) J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}{x \sqrt {x^k} \left (\text {Gamma}\left (\frac {2 b+k-1}{k}\right ) J_{\frac {2 b-1}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_1 \text {Gamma}\left (\frac {-2 b+k+1}{k}\right ) J_{\frac {1-2 b}{k}}\left (\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )} \\ y(x)\to \frac {\frac {k^2 \, _0\tilde {F}_1\left (;\frac {1-2 b}{k};-\frac {a x^k}{k^2}\right )-a x^k \, _0\tilde {F}_1\left (;\frac {1-2 b}{k}+2;-\frac {a x^k}{k^2}\right )}{k \, _0\tilde {F}_1\left (;\frac {-2 b+k+1}{k};-\frac {a x^k}{k^2}\right )}+1}{2 x} \\ y(x)\to \frac {\frac {k^2 \, _0\tilde {F}_1\left (;\frac {1-2 b}{k};-\frac {a x^k}{k^2}\right )-a x^k \, _0\tilde {F}_1\left (;\frac {1-2 b}{k}+2;-\frac {a x^k}{k^2}\right )}{k \, _0\tilde {F}_1\left (;\frac {-2 b+k+1}{k};-\frac {a x^k}{k^2}\right )}+1}{2 x} \\ \end{align*}