Internal problem ID [9637]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 50.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (a \,x^{2}+x b \right ) y-\alpha \,x^{2}-\beta x -\gamma =0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 724
dsolve(x^2*diff(y(x),x)=c*x^2*y(x)^2+(a*x^2+b*x)*y(x)+alpha*x^2+beta*x+gamma,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\sqrt {a^{2}-4 \alpha c}\, c_{1} a^{3} x -4 \sqrt {a^{2}-4 \alpha c}\, c_{1} a \alpha c x +c_{1} a^{4} x -8 c_{1} a^{2} \alpha c x +16 c_{1} \alpha ^{2} c^{2} x +\sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2} b -4 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha b c +c_{1} a^{3} b -2 c_{1} a^{2} \beta c -4 c_{1} a \alpha b c +8 c_{1} \alpha \beta \,c^{2}\right ) \WhittakerW \left (-\frac {b a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\left (-2 \sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2}+8 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha c \right ) \WhittakerW \left (\frac {-b a +2 \beta c +2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\left (\sqrt {a^{2}-4 \alpha c}\, a^{3} x -4 \sqrt {a^{2}-4 \alpha c}\, a \alpha c x +a^{4} x -8 a^{2} \alpha c x +16 \alpha ^{2} c^{2} x +\sqrt {a^{2}-4 \alpha c}\, a^{2} b -4 \sqrt {a^{2}-4 \alpha c}\, \alpha b c +a^{3} b -2 a^{2} \beta c -4 \alpha b c a +8 \alpha \beta \,c^{2}\right ) \WhittakerM \left (-\frac {b a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\left (\sqrt {a^{2}-4 \alpha c}\, \sqrt {b^{2}-4 c \gamma +2 b +1}\, a^{2}-4 \sqrt {a^{2}-4 \alpha c}\, \sqrt {b^{2}-4 c \gamma +2 b +1}\, \alpha c -a^{3} b +2 a^{2} \beta c +4 \alpha b c a -8 \alpha \beta \,c^{2}+\sqrt {a^{2}-4 \alpha c}\, a^{2}-4 \sqrt {a^{2}-4 \alpha c}\, \alpha c \right ) \WhittakerM \left (\frac {-b a +2 \beta c +2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )}{2 \left (a^{2}-4 \alpha c \right )^{\frac {3}{2}} x c \left (\WhittakerW \left (-\frac {b a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1}+\WhittakerM \left (-\frac {b a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.919 (sec). Leaf size: 1312
DSolve[x^2*y'[x]==c*x^2*y[x]^2+(a*x^2+b*x)*y[x]+\[Alpha]*x^2+\[Beta]*x+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\left (b+a x-x \sqrt {a^2-4 c \alpha }+\sqrt {b^2+2 b-4 c \gamma +1}+1\right ) c_1 \text {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )-x \left (a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )\right ) c_1 \text {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+2,x \sqrt {a^2-4 c \alpha }\right )+b L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+a x L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )-x \sqrt {a^2-4 c \alpha } L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+\sqrt {b^2+2 b-4 c \gamma +1} L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )-2 x \sqrt {a^2-4 c \alpha } L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}+1}\left (x \sqrt {a^2-4 c \alpha }\right )}{2 c x \left (c_1 \text {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )+L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )\right )} \\ y(x)\to \frac {\frac {\left (\sqrt {a^2-4 \alpha c} \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )+a b-2 \beta c\right ) \text {HypergeometricU}\left (\frac {\sqrt {a^2-4 \alpha c} \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )+a b-2 \beta c}{2 \sqrt {a^2-4 \alpha c}},\sqrt {b^2+2 b-4 c \gamma +1}+2,x \sqrt {a^2-4 \alpha c}\right )}{\text {HypergeometricU}\left (\frac {\sqrt {a^2-4 \alpha c} \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )+a b-2 \beta c}{2 \sqrt {a^2-4 \alpha c}},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 \alpha c}\right )}-\frac {-x \sqrt {a^2-4 \alpha c}+a x+\sqrt {b^2+2 b-4 c \gamma +1}+b+1}{x}}{2 c} \\ \end{align*}