Internal problem ID [3017]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 10
Problem number: 269.
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 y-c \,x^{4} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(x^2*diff(y(x),x) = a+b*x*y(x)+c*x^4*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {a c}\, c_{1} \BesselY \left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )}{x^{2} c \left (\BesselY \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1}+\BesselJ \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )}+\frac {\sqrt {a c}\, \BesselJ \left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )}{x^{2} c \left (\BesselY \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1}+\BesselJ \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.367 (sec). Leaf size: 225
DSolve[x^2 y'[x]==a+b x y[x]+c x^4 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \left (\sec \left (\frac {\pi b}{2}\right ) J_{\frac {1}{2} (-b-1)}\left (\sqrt {a} \sqrt {c} x\right )+\left (\tan \left (\frac {\pi b}{2}\right )-c_1\right ) J_{\frac {b+1}{2}}\left (\sqrt {a} \sqrt {c} x\right )\right )}{\sqrt {c} x^2 \left (Y_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )+c_1 J_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )\right )} \\ y(x)\to -\frac {2 \, _0\tilde {F}_1\left (;\frac {b+3}{2};-\frac {1}{4} a c x^2\right )}{c x^3 \, _0\tilde {F}_1\left (;\frac {b+5}{2};-\frac {1}{4} a c x^2\right )} \\ y(x)\to -\frac {2 \, _0\tilde {F}_1\left (;\frac {b+3}{2};-\frac {1}{4} a c x^2\right )}{c x^3 \, _0\tilde {F}_1\left (;\frac {b+5}{2};-\frac {1}{4} a c x^2\right )} \\ \end{align*}