61.2.69 problem 69

Internal problem ID [11996]
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
Date solved : Sunday, March 30, 2025 at 10:06:58 PM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 58

ode:=(a*x^2+b*x+e)*(-y(x)+x*diff(y(x),x))-y(x)^2+x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\tanh \left (\frac {c_1 \sqrt {4 e a -b^{2}}+2 \arctan \left (\frac {2 a x +b}{\sqrt {4 e a -b^{2}}}\right )}{\sqrt {4 e a -b^{2}}}\right ) x \]

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 53

ode=(a*x^2+b*x+e)*(x*D[y[x],x]-y[x])-y[x]^2+x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(K[1]-1) (K[1]+1)}dK[1]=\int _1^x\frac {1}{a K[2]^2+b K[2]+e}dK[2]+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
e = symbols("e") 
y = Function("y") 
ode = Eq(x**2 + (x*Derivative(y(x), x) - y(x))*(a*x**2 + b*x + e) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out