Internal problem ID [10388]
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]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 x y=-\left (1-a \right ) x^{2}+b} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
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)
\[ y \left (x \right ) = x +\frac {\sqrt {a b}}{c_{1} \sqrt {a b}+\arctan \left (\frac {a x}{\sqrt {a b}}\right )} \]
✓ Solution by Mathematica
Time used: 0.562 (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 {\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}+c_1} \\ y(x)\to x \\ \end{align*}