Internal problem ID [9655]
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: 68.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} \left (a +x \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (x b +c \right ) y+\alpha x +\beta =0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 5852
dsolve(x^2*(x+a)*(diff(y(x),x)+lambda*y(x)^2)+x*(b*x+c)*y(x)+alpha*x+beta=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.837 (sec). Leaf size: 1545
DSolve[x^2*(x+a)*(y'[x]+\[Lambda]*y[x]^2)+x*(b*x+c)*y[x]+\[Alpha]*x+\[Beta]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \left (\frac {2 x \left ((b+2 \alpha \lambda ) a^2-\left (c b+\sqrt {(a-c)^2-4 a \beta \lambda } b+c+2 \beta \lambda \right ) a+c \left (c+\sqrt {(a-c)^2-4 a \beta \lambda }\right )\right ) \, _2F_1\left (-\frac {c+a \left (-b+\sqrt {(b-1)^2-4 \alpha \lambda }-2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},-\frac {c-a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }+2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};2-\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )}{\sqrt {(a-c)^2-4 a \beta \lambda }-a}-2 a \left (-a+c+\sqrt {(a-c)^2-4 a \beta \lambda }\right ) \, _2F_1\left (-\frac {c+a \left (\sqrt {(b-1)^2-4 \alpha \lambda }-b\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},-\frac {c-a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {a-\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )\right ) a^{\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a}}+x^{\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a}} \left (2 a \left (a-c+\sqrt {(a-c)^2-4 a \beta \lambda }\right ) \, _2F_1\left (\frac {-c+a \left (b-\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},\frac {-c+a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {a+\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )-\frac {2 x \left ((b+2 \alpha \lambda ) a^2-(b c+c+2 \beta \lambda ) a+b \sqrt {(a-c)^2-4 a \beta \lambda } a+c^2-c \sqrt {(a-c)^2-4 a \beta \lambda }\right ) \, _2F_1\left (\frac {-c+a \left (b-\sqrt {(b-1)^2-4 \alpha \lambda }+2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},\frac {-c+a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }+2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a}+2;-\frac {x}{a}\right )}{a+\sqrt {(a-c)^2-4 a \beta \lambda }}\right )}{4 a^2 x \lambda \left (c_1 \, _2F_1\left (-\frac {c+a \left (\sqrt {(b-1)^2-4 \alpha \lambda }-b\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},-\frac {c-a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {a-\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right ) a^{\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a}}+x^{\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a}} \, _2F_1\left (\frac {-c+a \left (b-\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},\frac {-c+a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {a+\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )\right )} \\ y(x)\to \frac {\frac {\left (-2 \alpha a^3 \lambda -a^2 \left (2 \alpha \lambda \sqrt {(a-c)^2-4 a \beta \lambda }+4 b \beta \lambda +b c-2 \beta \lambda \right )+a \left (b c \sqrt {(a-c)^2-4 a \beta \lambda }+2 \beta \lambda \sqrt {(a-c)^2-4 a \beta \lambda }+c (b c+4 \beta \lambda +c)\right )-c^2 \left (\sqrt {(a-c)^2-4 a \beta \lambda }+c\right )\right ) \, _2F_1\left (-\frac {c+a \left (-b+\sqrt {(b-1)^2-4 \alpha \lambda }-2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},-\frac {c-a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }+2\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};2-\frac {\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )}{\, _2F_1\left (-\frac {c+a \left (\sqrt {(b-1)^2-4 \alpha \lambda }-b\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a},-\frac {c-a \left (b+\sqrt {(b-1)^2-4 \alpha \lambda }\right )+\sqrt {(a-c)^2-4 a \beta \lambda }}{2 a};\frac {a-\sqrt {(a-c)^2-4 a \beta \lambda }}{a};-\frac {x}{a}\right )}+\frac {a \left (c^2-2 a (2 \beta \lambda +c)\right ) \left (\sqrt {(a-c)^2-4 a \beta \lambda }-a+c\right )}{x}}{2 a^2 \lambda \left (2 a (2 \beta \lambda +c)-c^2\right )} \\ \end{align*}