Internal problem ID [9645]
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: 58.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 y x +\left (-a +1\right ) x^{2}-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve((a*x^2+b)*diff(y(x),x)+y(x)^2-2*x*y(x)+(1-a)*x^2-b=0,y(x), singsol=all)
\[ y \relax (x ) = x +\frac {1}{c_{1}+\frac {\arctan \left (\frac {a x}{\sqrt {b a}}\right )}{\sqrt {b a}}} \]
✓ Solution by Mathematica
Time used: 0.382 (sec). Leaf size: 41
DSolve[(a*x^2+b)*y'[x]+y[x]^2-2*x*y[x]+(1-a)*x^2-b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\frac {1}{\frac {\text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}+c_1} \\ y(x)\to x \\ \end{align*}