Internal problem ID [9654]
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: 67.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 1348
dsolve(x*(x^2+a)*(diff(y(x),x)+lambda*y(x)^2)+(b*x^2+c)*y(x)+s*x=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.459 (sec). Leaf size: 783
DSolve[x*(x^2+a)*(y'[x]+\[Lambda]*y[x]^2)+(b*x^2+c)*y[x]+s*x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {a^{\frac {1}{2} \left (\frac {c}{a}-5\right )} x^{-\frac {c}{a}} \left (a \left (3 a^2-4 a c+c^2\right ) \, _2F_1\left (\frac {a \left (b-\sqrt {(b-1)^2-4 s \lambda }+1\right )-2 c}{4 a},\frac {a \left (b+\sqrt {(b-1)^2-4 s \lambda }+1\right )-2 c}{4 a};\frac {3}{2}-\frac {c}{2 a};-\frac {x^2}{a}\right )-x^2 \left (a^2 (b+\lambda s)-a (b+1) c+c^2\right ) \, _2F_1\left (\frac {a \left (b-\sqrt {(b-1)^2-4 s \lambda }+5\right )-2 c}{4 a},\frac {a \left (b+\sqrt {(b-1)^2-4 s \lambda }+5\right )-2 c}{4 a};\frac {5}{2}-\frac {c}{2 a};-\frac {x^2}{a}\right )\right )}{3 a-c}-\frac {c_1 \lambda s x \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }+3\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }+3\right );\frac {1}{2} \left (\frac {c}{a}+3\right );-\frac {x^2}{a}\right )}{a+c}}{\lambda a^{\frac {1}{2} \left (\frac {c}{a}-1\right )} x^{1-\frac {c}{a}} \, _2F_1\left (\frac {a \left (b-\sqrt {(b-1)^2-4 s \lambda }+1\right )-2 c}{4 a},\frac {a \left (b+\sqrt {(b-1)^2-4 s \lambda }+1\right )-2 c}{4 a};\frac {3}{2}-\frac {c}{2 a};-\frac {x^2}{a}\right )+c_1 \lambda \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }-1\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }-1\right );\frac {a+c}{2 a};-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }+3\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }+3\right );\frac {1}{2} \left (\frac {c}{a}+3\right );-\frac {x^2}{a}\right )}{(a+c) \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }-1\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }-1\right );\frac {a+c}{2 a};-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }+3\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }+3\right );\frac {1}{2} \left (\frac {c}{a}+3\right );-\frac {x^2}{a}\right )}{(a+c) \, _2F_1\left (\frac {1}{4} \left (b-\sqrt {(b-1)^2-4 s \lambda }-1\right ),\frac {1}{4} \left (b+\sqrt {(b-1)^2-4 s \lambda }-1\right );\frac {a+c}{2 a};-\frac {x^2}{a}\right )} \\ \end{align*}