Internal problem ID [10396]
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: 66.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-x \left (b x +c \right ) y=\alpha x +\beta } \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 438
dsolve(x^3*diff(y(x),x)=a*x^3*y(x)^2+x*(b*x+c)*y(x)+alpha*x+beta,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+x \left (b +2\right ) c -a x \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c_{1} \left (-c^{2}-x \left (b +2\right ) c +a x \beta \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+x \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) \left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right )\right )}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.395 (sec). Leaf size: 908
DSolve[x^3*y'[x]==a*x^3*y[x]^2+x*(b*x+c)*y[x]+\[Alpha]*x+\[Beta],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\frac {c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}+1} \left (c \left (\sqrt {-4 a \alpha +b^2+2 b+1}-b-1\right )+2 a \beta \right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}+1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+2,-\frac {c}{x}\right )}{\sqrt {-4 a \alpha +b^2+2 b+1}+1}-\left (\sqrt {-4 a \alpha +b^2+2 b+1}-b-1\right ) c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}-1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+1,-\frac {c}{x}\right )+\frac {c_1 \left (\left (c \left (\sqrt {-4 a \alpha +b^2+2 b+1}+b+1\right )-2 a \beta \right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}-1\right )}{2 c},2-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )+x \left (-4 a \alpha +b \sqrt {-4 a \alpha +b^2+2 b+1}+b^2+b\right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )\right )}{x \left (\sqrt {-4 a \alpha +b^2+2 b+1}-1\right )}}{2 a x \left (c^{\sqrt {-4 a \alpha +b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a \alpha +b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {b^2+2 b-4 a \alpha +1}-1\right ),\sqrt {b^2+2 b-4 a \alpha +1}+1,-\frac {c}{x}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )\right )} \\ y(x)\to \frac {\frac {\left (c \left (b \left (\sqrt {-4 a \alpha +b^2+2 b+1}+3\right )+2 \left (-2 a \alpha +\sqrt {-4 a \alpha +b^2+2 b+1}+1\right )+b^2\right )-2 a \beta \left (\sqrt {-4 a \alpha +b^2+2 b+1}+1\right )\right ) \operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}-1\right )}{2 c},2-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )}{\operatorname {Hypergeometric1F1}\left (\frac {2 a \beta -c \left (b+\sqrt {b^2+2 b-4 a \alpha +1}+1\right )}{2 c},1-\sqrt {b^2+2 b-4 a \alpha +1},-\frac {c}{x}\right )}+x \left (-4 a \alpha +b^2+2 b\right ) \left (\sqrt {-4 a \alpha +b^2+2 b+1}+b+1\right )}{2 a x^2 \left (4 a \alpha -b^2-2 b\right )} \\ \end{align*}