61.2.58 problem 58

Internal problem ID [12064]
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
Date solved : Tuesday, January 28, 2025 at 12:10:18 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 y x +\left (1-a \right ) x^{2}-b&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (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)
 

\[ y = x +\frac {\sqrt {a b}}{c_{1} \sqrt {a b}+\arctan \left (\frac {a x}{\sqrt {a b}}\right )} \]

✓ Solution by Mathematica

Time used: 0.372 (sec). Leaf size: 38

DSolve[(a*x^2+b)*D[y[x],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}{-\int _1^x-\frac {1}{a K[1]^2+b}dK[1]+c_1} \\ y(x)\to x \\ \end{align*}