29.9.25 problem 265

Internal problem ID [4865]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 265
Date solved : Sunday, March 30, 2025 at 04:07:49 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 52

ode:=x^2*diff(y(x),x)+2+a*x*(1-x*y(x))-x^2*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+c_1}{\left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x}+c_1 \right ) x} \]

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 78

ode=x^2 D[y[x],x]+2+a x(1-x y[x])-x^2 y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {e^{a x} \left (-a^3 x^3+a^2 x^2-2 a x+2\right )+a^3 c_1}{x \left (e^{a x} \left (a^2 x^2-2 a x+2\right )+a^3 c_1\right )} \\ y(x)\to \frac {1}{x} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*x*(-x*y(x) + 1) - x**2*y(x)**2 + x**2*Derivative(y(x), x) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x) + a/x - y(x)**2 + Derivative(y(x), x) + 2/x**2 cannot be solved by the factorable group method