Internal problem ID [9600]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, [_Riccati, _special]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 48
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\tan \left (-\frac {\ln \relax (x ) \sqrt {4 a b -1}}{2}+\frac {c_{1} \sqrt {4 a b -1}}{2}\right ) \sqrt {4 a b -1}+1}{2 a x} \]
✓ Solution by Mathematica
Time used: 0.303 (sec). Leaf size: 77
DSolve[x^2*y'[x]==a*x^2*y[x]^2+b,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-1+\sqrt {1-4 a b} \left (-1+\frac {2 c_1}{x^{\sqrt {1-4 a b}}+c_1}\right )}{2 a x} \\ y(x)\to \frac {\sqrt {1-4 a b}-1}{2 a x} \\ \end{align*}