Internal problem ID [3011]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 9
Problem number: 263.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x^{2}-a -b \,x^{n}-x^{2} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 275
dsolve(x^2*diff(y(x),x) = a+b*x^n+x^2*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {x^{\frac {n}{2}} \sqrt {b}\, c_{1} \BesselY \left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )}{x \left (\BesselY \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )}+\frac {\left (-\sqrt {1-4 a}\, c_{1}-c_{1}\right ) \BesselY \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )+2 \BesselJ \left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {b}\, x^{\frac {n}{2}}+\left (-\sqrt {1-4 a}-1\right ) \BesselJ \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )}{2 x \left (\BesselY \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.785 (sec). Leaf size: 807
DSolve[x^2 y'[x]==a+b x^n + x^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} b^{\frac {i \sqrt {4 a-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}} \text {Gamma}\left (\frac {\sqrt {1-4 a}+n}{n}\right ) \left (2 b x^n \left (\frac {\sqrt {b} \sqrt [3]{\left (x^n\right )^{3/2}}}{n}\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \, _0\tilde {F}_1\left (;\frac {\sqrt {(1-4 a) n^2}}{n^2}+2;-\frac {b x^n}{n^2}\right )-i \left (\sqrt {4 a-1}-i\right ) n \left (\frac {\sqrt {b} \sqrt {x^n}}{n}\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \, _0\tilde {F}_1\left (;\frac {\sqrt {(1-4 a) n^2}}{n^2}+1;-\frac {b x^n}{n^2}\right )\right )+c_1 n^{\frac {2 i \sqrt {4 a-1}}{n}} b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \text {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) \left (2 b x^n \left (\frac {\sqrt {b} \sqrt [3]{\left (x^n\right )^{3/2}}}{n}\right )^{-\frac {\sqrt {(1-4 a) n^2}}{n^2}} \, _0\tilde {F}_1\left (;2-\frac {\sqrt {(1-4 a) n^2}}{n^2};-\frac {b x^n}{n^2}\right )+i \left (\sqrt {4 a-1}+i\right ) n \left (\frac {\sqrt {b} \sqrt {x^n}}{n}\right )^{-\frac {\sqrt {(1-4 a) n^2}}{n^2}} \, _0\tilde {F}_1\left (;1-\frac {\sqrt {(1-4 a) n^2}}{n^2};-\frac {b x^n}{n^2}\right )\right )}{2 n x \left (n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} b^{\frac {i \sqrt {4 a-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}} \text {Gamma}\left (\frac {\sqrt {1-4 a}+n}{n}\right ) J_{\frac {\sqrt {(1-4 a) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )+c_1 n^{\frac {2 i \sqrt {4 a-1}}{n}} b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} \text {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) J_{-\frac {\sqrt {(1-4 a) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )\right )} \\ y(x)\to \frac {\frac {2 b x^n \, _0\tilde {F}_1\left (;2-\frac {\sqrt {(1-4 a) n^2}}{n^2};-\frac {b x^n}{n^2}\right )}{n \, _0\tilde {F}_1\left (;1-\frac {\sqrt {(1-4 a) n^2}}{n^2};-\frac {b x^n}{n^2}\right )}+i \sqrt {4 a-1}-1}{2 x} \\ \end{align*}