60.1.139 problem 142

Internal problem ID [10153]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 142
Date solved : Sunday, March 30, 2025 at 03:20:18 PM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

ode:=x^2*(diff(y(x),x)-y(x)^2)-y(x)*a*x^2+a*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: 0.852 (sec). Leaf size: 76

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

\begin{align*} y(x)\to \frac {\int _1^xe^{a K[1]+2} K[1]^2dK[1]+x^3 \left (-e^{a x+2}\right )+c_1}{x \left (\int _1^xe^{a K[1]+2} K[1]^2dK[1]+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**2*y(x) + a*x + x**2*(-y(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