Internal problem ID [9636]
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: 49.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b x y-c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 73
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b*x*y(x)+c,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\tan \left (-\frac {\ln \relax (x ) \sqrt {4 a c -b^{2}-2 b -1}}{2}+\frac {c_{1} \sqrt {4 a c -b^{2}-2 b -1}}{2}\right ) \sqrt {4 a c -b^{2}-2 b -1}+b +1}{2 a x} \]
✓ Solution by Mathematica
Time used: 0.439 (sec). Leaf size: 93
DSolve[x^2*y'[x]==a*x^2*y[x]^2+b*x*y[x]+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {(b+1)^2-4 a c} \left (1-\frac {2 c_1}{x^{\sqrt {(b+1)^2-4 a c}}+c_1}\right )+b+1}{2 a x} \\ y(x)\to \frac {\sqrt {(b+1)^2-4 a c}-b-1}{2 a x} \\ \end{align*}