Internal problem ID [9653]
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]
Solve \begin {gather*} \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-x \left (x b +c \right ) y-\alpha x -\beta =0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 615
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)
\[ y \relax (x ) = \frac {\left (2 a^{2} \beta ^{2} x c_{1}+2 a \alpha \,c^{2} x c_{1}-2 a b \beta c x c_{1}-6 a \beta c x c_{1}+2 b \,c^{2} x c_{1}+4 c^{2} x c_{1}\right ) \KummerU \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +5 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{2 x^{2} c^{2} a \left (\KummerU \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1}+\KummerM \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )}+\frac {\left (2 a \beta c x c_{1}-2 b \,c^{2} x c_{1}-2 c^{3} c_{1}-4 c^{2} x c_{1}\right ) \KummerU \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c^{2} x -2 a \beta c x +b \,c^{2} x +3 c^{2} x \right ) \KummerM \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +5 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (2 a \beta c x -2 b \,c^{2} x -2 c^{3}-4 c^{2} x \right ) \KummerM \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{2 x^{2} c^{2} a \left (\KummerU \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1}+\KummerM \left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c -2 a \beta +b c +3 c}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.257 (sec). Leaf size: 806
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 {\Gamma \left (\sqrt {(b+1)^2-4 a \alpha }+1\right ) c^{\sqrt {(b+1)^2-4 a \alpha }} \left (\frac {1}{x}\right )^{\sqrt {(b+1)^2-4 a \alpha }} \left (\left (-\sqrt {(b+1)^2-4 a \alpha }+b+1\right ) \, _1\tilde {F}_1\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {(b+1)^2-4 a \alpha }-1\right );\sqrt {(b+1)^2-4 a \alpha }+1;-\frac {c}{x}\right )+\frac {\left (c \left (\sqrt {(b+1)^2-4 a \alpha }-b-1\right )+2 a \beta \right ) \, _1\tilde {F}_1\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {(b+1)^2-4 a \alpha }+1\right );\sqrt {(b+1)^2-4 a \alpha }+2;-\frac {c}{x}\right )}{x}\right )-\frac {c_1 \left (\frac {\left (c \left (\sqrt {(b+1)^2-4 a \alpha }+b+1\right )-2 a \beta \right ) \, _1F_1\left (\frac {2 a \beta -c \left (b+\sqrt {(b+1)^2-4 a \alpha }-1\right )}{2 c};2-\sqrt {(b+1)^2-4 a \alpha };-\frac {c}{x}\right )}{x}+\left (b \left (\sqrt {(b+1)^2-4 a \alpha }+b+1\right )-4 a \alpha \right ) \, _1F_1\left (\frac {2 a \beta -c \left (b+\sqrt {(b+1)^2-4 a \alpha }+1\right )}{2 c};1-\sqrt {(b+1)^2-4 a \alpha };-\frac {c}{x}\right )\right )}{1-\sqrt {(b+1)^2-4 a \alpha }}}{2 a x \left (c^{\sqrt {(b+1)^2-4 a \alpha }} \left (\frac {1}{x}\right )^{\sqrt {(b+1)^2-4 a \alpha }} \, _1F_1\left (\frac {1}{2} \left (-b+\frac {2 a \beta }{c}+\sqrt {(b+1)^2-4 a \alpha }-1\right );\sqrt {(b+1)^2-4 a \alpha }+1;-\frac {c}{x}\right )+c_1 \, _1F_1\left (\frac {2 a \beta -c \left (b+\sqrt {(b+1)^2-4 a \alpha }+1\right )}{2 c};1-\sqrt {(b+1)^2-4 a \alpha };-\frac {c}{x}\right )\right )} \\ y(x)\to \frac {\frac {\left (-2 a \beta \left (\sqrt {(b+1)^2-4 a \alpha }+1\right )+b c \left (\sqrt {(b+1)^2-4 a \alpha }+b+3\right )+2 c \left (-2 a \alpha +\sqrt {(b+1)^2-4 a \alpha }+1\right )\right ) \, _1F_1\left (\frac {2 a \beta -c \left (b+\sqrt {(b+1)^2-4 a \alpha }-1\right )}{2 c};2-\sqrt {(b+1)^2-4 a \alpha };-\frac {c}{x}\right )}{\, _1F_1\left (\frac {2 a \beta -c \left (b+\sqrt {(b+1)^2-4 a \alpha }+1\right )}{2 c};1-\sqrt {(b+1)^2-4 a \alpha };-\frac {c}{x}\right )}+x (b (b+2)-4 a \alpha ) \left (\sqrt {(b+1)^2-4 a \alpha }+b+1\right )}{2 a x^2 (4 a \alpha -b (b+2))} \\ \end{align*}