Internal problem ID [10398]
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]
\[ \boxed {x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y=-\alpha x -\beta } \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1508
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: 5.239 (sec). Leaf size: 1770
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 {2 a \left (a-c+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {-c+a \left (b-\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {-c+a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {a+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right ) x^{\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}}-\frac {\left (-c+a \left (b-\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \left (-c+a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {-c+a \left (b-\sqrt {b^2-2 b-4 \alpha \lambda +1}+2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {-c+a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}+2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}+2,-\frac {x}{a}\right ) x^{\frac {a+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}}}{a+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}+2 a^{\frac {a+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}} c_1 \left (\frac {\left (c+a \left (\sqrt {b^2-2 b-4 \alpha \lambda +1}-b\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \left (c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {c+a \left (-b+\sqrt {b^2-2 b-4 \alpha \lambda +1}-2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},-\frac {c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}+2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},2-\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right )}{2 a \left (\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}-a\right )}-\frac {\left (-a+c+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {c+a \left (\sqrt {b^2-2 b-4 \alpha \lambda +1}-b\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},-\frac {c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},1-\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right )}{x}\right ) x}{4 a^2 x \lambda \left (c_1 \operatorname {Hypergeometric2F1}\left (-\frac {c+a \left (\sqrt {b^2-2 b-4 \alpha \lambda +1}-b\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},-\frac {c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},1-\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right ) a^{\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}}+x^{\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {-c+a \left (b-\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {-c+a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},\frac {a+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right )\right )} \\ y(x)\to \frac {\frac {a \left (c^2-2 a (2 \beta \lambda +c)\right ) \left (\sqrt {a^2-2 a (2 \beta \lambda +c)+c^2}-a+c\right )}{x}-\frac {\left (2 \alpha a^3 \lambda +a^2 \left (2 \alpha \lambda \sqrt {a^2-2 a (2 \beta \lambda +c)+c^2}+4 b \beta \lambda +b c-2 \beta \lambda \right )-a \left (b c \sqrt {a^2-2 a (2 \beta \lambda +c)+c^2}+2 \beta \lambda \sqrt {a^2-2 a (2 \beta \lambda +c)+c^2}+b c^2+c^2+4 \beta c \lambda \right )+c^2 \left (\sqrt {a^2-2 a (2 \beta \lambda +c)+c^2}+c\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {c+a \left (-b+\sqrt {b^2-2 b-4 \alpha \lambda +1}-2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},-\frac {c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}+2\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},2-\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right )}{\operatorname {Hypergeometric2F1}\left (-\frac {c+a \left (\sqrt {b^2-2 b-4 \alpha \lambda +1}-b\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},-\frac {c-a \left (b+\sqrt {b^2-2 b-4 \alpha \lambda +1}\right )+\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{2 a},1-\frac {\sqrt {a^2-2 (c+2 \beta \lambda ) a+c^2}}{a},-\frac {x}{a}\right )}}{2 a^2 \lambda \left (2 a (2 \beta \lambda +c)-c^2\right )} \\ \end{align*}