61.2.50 problem 50

Internal problem ID [12056]
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
Date solved : Monday, January 27, 2025 at 11:55:02 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{2} y^{\prime }&=c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 443

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)
 

\[ y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-a b +2 \beta c +\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (a x +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 c \left (\alpha x +\frac {\beta }{2}\right )\right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) c x} \]

✓ Solution by Mathematica

Time used: 4.143 (sec). Leaf size: 1312

DSolve[x^2*D[y[x],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 \operatorname {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 \operatorname {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 \operatorname {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 ) \operatorname {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 )}{\operatorname {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 )}-\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*}