Internal problem ID [3525]
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]
\[ \boxed {x^{2} y^{\prime }-b x y-c \,x^{4} y^{2}=a} \]
✓ 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)
\[ y \left (x \right ) = \frac {\sqrt {a c}\, c_{1} \operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )}{x^{2} c \left (\operatorname {BesselY}\left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselJ}\left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )}+\frac {\sqrt {a c}\, \operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )}{x^{2} c \left (\operatorname {BesselY}\left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right ) c_{1} +\operatorname {BesselJ}\left (-\frac {3}{2}-\frac {b}{2}, \sqrt {a c}\, x \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.387 (sec). Leaf size: 394
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} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )+\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+b c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+3 c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \left (\operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )\right )} y(x)\to -\frac {\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )} \end{align*}