Internal problem ID [10410]
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: 70.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y=-s} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 5730
dsolve(x^2*(x^2+a)*(diff(y(x),x)+lambda*y(x)^2)+x*(b*x^2+c)*y(x)+s=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 7.158 (sec). Leaf size: 1470
DSolve[x^2*(x^2+a)*(y'[x]+\[Lambda]*y[x]^2)+x*(b*x^2+c)*y[x]+s==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {\left (a-c-\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) c_1 \left (\left (-2 b a+a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {-5 a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},-\frac {-a (2 b+3)+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},2-\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a},-\frac {x^2}{a}\right ) x^2+2 a \left (2 a-\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a},-\frac {x^2}{a}\right )\right ) a^{\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a}}}{2 a-\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}+2 x^{\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{a}} \left (a-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {-2 b a+a+c-\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {a-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a}+1,-\frac {x^2}{a}\right ) a-\frac {x^{\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{a}+2} \left (a-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) \left (a (2 b-1)-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5 a-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {a (2 b+3)-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a}+2,-\frac {x^2}{a}\right )}{2 a+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}}{4 a^2 x \lambda \left (c_1 \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a},-\frac {x^2}{a}\right ) a^{\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a}}+x^{\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {-2 b a+a+c-\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {a-c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a}+1,-\frac {x^2}{a}\right )\right )} y(x)\to \frac {x \left (a^3 (-b)+a^2 \left (b \sqrt {a^2-2 a (c+2 \lambda s)+c^2}-4 (b-1) \lambda s+c\right )+a \left (b c \left (\sqrt {a^2-2 a (c+2 \lambda s)+c^2}+c\right )-c \sqrt {a^2-2 a (c+2 \lambda s)+c^2}+2 \lambda s \sqrt {a^2-2 a (c+2 \lambda s)+c^2}+4 c \lambda s\right )-c^2 \left (\sqrt {a^2-2 a (c+2 \lambda s)+c^2}+c\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {-5 a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},-\frac {-a (2 b+3)+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},2-\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a},-\frac {x^2}{a}\right )}{2 a^2 \lambda \left (3 a^2+2 a (c+2 \lambda s)-c^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 s \lambda ) a+c^2}}{2 a},-\frac {x^2}{a}\right )}-\frac {\sqrt {a^2-2 a (c+2 \lambda s)+c^2}-a+c}{2 a \lambda x} \end{align*}