Internal problem ID [9632]
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]
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b x y-c=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 57
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b*x*y(x)+c,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {b +1+\tan \left (\frac {\sqrt {4 a c -b^{2}-2 b -1}\, \left (-\ln \left (x \right )+c_{1} \right )}{2}\right ) \sqrt {4 a c -b^{2}-2 b -1}}{2 a x} \]
✓ Solution by Mathematica
Time used: 0.29 (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*}