Internal problem ID [10383]
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: 53.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} y^{\prime }-c \,x^{2} y^{2}-\left (a \,x^{n}+b \right ) x y=\gamma +\beta \,x^{n}+\alpha \,x^{2 n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 560
dsolve(x^2*diff(y(x),x)=c*x^2*y(x)^2+(a*x^n+b)*x*y(x)+alpha*x^(2*n)+beta*x^n+gamma,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}+\sqrt {a^{2}-4 \alpha c}\, n +\left (n -b -1\right ) a +2 \beta c \right ) \operatorname {WhittakerM}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )-2 \operatorname {WhittakerW}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} n \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) \left (\left (a \,x^{n}+b -n +1\right ) \sqrt {a^{2}-4 \alpha c}+\left (a^{2}-4 \alpha c \right ) x^{n}+a \left (b -n +1\right )-2 \beta c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 3.672 (sec). Leaf size: 1837
DSolve[x^2*y'[x]==c*x^2*y[x]^2+(a*x^n+b)*x*y[x]+\[Alpha]*x^(2*n)+\[Beta]*x^n+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\left (\left (-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) c_1 \operatorname {HypergeometricU}\left (\frac {\left (3 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (6 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {2 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right ) x^n\right )-n \left (-a^2 n x^n+4 c n \alpha x^n+a n \sqrt {a^2-4 c \alpha } x^n+\sqrt {a^2-4 c \alpha } \left (b n+n+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) c_1 \operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )-n \left (\left (-a^2 n x^n+4 c n \alpha x^n+a n \sqrt {a^2-4 c \alpha } x^n+\sqrt {a^2-4 c \alpha } \left (b n+n+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )-2 n x^n \left (a^2-4 c \alpha \right ) L_{\frac {-\left (\left (3 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (6 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )\right )}{2 c n^2 x \sqrt {a^2-4 c \alpha } \left (c_1 \operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )+L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )\right )} \\ y(x)\to \frac {-\frac {x^n \left (2 c \left (\beta n \sqrt {a^2-4 \alpha c}+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+2 \alpha n^2\right )-\left (a^2 \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+n^2\right )\right )+a n (-b+n-1) \sqrt {a^2-4 \alpha c}\right ) \operatorname {HypergeometricU}\left (\frac {\left (3 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (6 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {2 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )}{\operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )}-n \left (\sqrt {a^2-4 \alpha c} \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+b n+n\right )+a n x^n \sqrt {a^2-4 \alpha c}-a^2 n x^n+4 \alpha c n x^n\right )}{2 c n^2 x \sqrt {a^2-4 \alpha c}} \\ \end{align*}