Internal problem ID [10409]
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: 69.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class D`], _rational, _Riccati]
\[ \boxed {\left (a \,x^{2}+b x +e \right ) \left (y^{\prime } x -y\right )-y^{2}=-x^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 58
dsolve((a*x^2+b*x+e)*(x*diff(y(x),x)-y(x))-y(x)^2+x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = -\tanh \left (\frac {c_{1} \sqrt {4 e a -b^{2}}+2 \arctan \left (\frac {2 a x +b}{\sqrt {4 e a -b^{2}}}\right )}{\sqrt {4 e a -b^{2}}}\right ) x \]
✓ Solution by Mathematica
Time used: 1.973 (sec). Leaf size: 116
DSolve[(a*x^2+b*x+e)*(x*y'[x]-y[x])-y[x]^2+x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \left (-1+\exp \left (\frac {4 \arctan \left (\frac {2 a x+b}{\sqrt {4 a e-b^2}}\right )}{\sqrt {4 a e-b^2}}+2 c_1\right )\right )}{1+\exp \left (\frac {4 \arctan \left (\frac {2 a x+b}{\sqrt {4 a e-b^2}}\right )}{\sqrt {4 a e-b^2}}+2 c_1\right )} y(x)\to -x y(x)\to x \end{align*}