Internal problem ID [9657]
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]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (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: 8.055 (sec). Leaf size: 1285
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 {\left (a-c-\sqrt {(a-c)^2-4 a s \lambda }\right ) c_1 \Gamma \left (1-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}\right ) \left (4 \, _2\tilde {F}_1\left (-\frac {-a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},-\frac {-2 b a+a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};1-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a};-\frac {x^2}{a}\right ) a^2+x^2 \left (-2 b a+a+c+\sqrt {(a-c)^2-4 a s \lambda }\right ) \, _2\tilde {F}_1\left (-\frac {-5 a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},-\frac {-a (2 b+3)+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};2-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a};-\frac {x^2}{a}\right )\right ) a^{\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}}+x^{\frac {\sqrt {(a-c)^2-4 a s \lambda }}{a}} \left (a-c+\sqrt {(a-c)^2-4 a s \lambda }\right ) \left (4 \, _2F_1\left (\frac {a-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},\frac {a (2 b-1)-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}+1;-\frac {x^2}{a}\right ) a^2+\frac {2 x^2 \left (-2 b a+a+c-\sqrt {(a-c)^2-4 a s \lambda }\right ) \, _2F_1\left (\frac {5 a-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},\frac {a (2 b+3)-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}+2;-\frac {x^2}{a}\right ) a}{2 a+\sqrt {(a-c)^2-4 a s \lambda }}\right )}{8 a^3 x \lambda \left (c_1 \, _2F_1\left (-\frac {-a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},-\frac {-2 b a+a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};1-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a};-\frac {x^2}{a}\right ) a^{\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}}+x^{\frac {\sqrt {(a-c)^2-4 a s \lambda }}{a}} \, _2F_1\left (\frac {a-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},\frac {a (2 b-1)-c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a}+1;-\frac {x^2}{a}\right )\right )} \\ y(x)\to \frac {x \left (a^3 (-b)+a^2 \left (b \sqrt {(a-c)^2-4 a \lambda s}-4 (b-1) \lambda s+c\right )+a \left (b c \left (\sqrt {(a-c)^2-4 a \lambda s}+c\right )-c \sqrt {(a-c)^2-4 a \lambda s}+2 \lambda s \sqrt {(a-c)^2-4 a \lambda s}+4 c \lambda s\right )-c^2 \left (\sqrt {(a-c)^2-4 a \lambda s}+c\right )\right ) \, _2F_1\left (-\frac {-5 a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},-\frac {-a (2 b+3)+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};2-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a};-\frac {x^2}{a}\right )}{2 a^2 \lambda \left (3 a^2+2 a (c+2 \lambda s)-c^2\right ) \, _2F_1\left (-\frac {-a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a},-\frac {-2 b a+a+c+\sqrt {(a-c)^2-4 a s \lambda }}{4 a};1-\frac {\sqrt {(a-c)^2-4 a s \lambda }}{2 a};-\frac {x^2}{a}\right )}-\frac {\sqrt {(a-c)^2-4 a \lambda s}-a+c}{2 a \lambda x} \\ \end{align*}