47.2.51 problem 47

Internal problem ID [7467]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 47
Date solved : Wednesday, March 05, 2025 at 04:39:03 AM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 2 x y^{\prime }+y&=y^{2} \sqrt {x -x^{2} y^{2}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 38

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

\[ -\frac {-1+x y^{2}}{y \sqrt {-x \left (-1+x y^{2}\right )}}+\frac {\ln \left (x \right )}{2}-c_{1} = 0 \]

✓ Mathematica. Time used: 0.755 (sec). Leaf size: 62

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

\begin{align*} y(x)\to -\frac {2}{\sqrt {x \left (\log ^2(x)-2 c_1 \log (x)+4+c_1{}^2\right )}} \\ y(x)\to \frac {2}{\sqrt {x \left (\log ^2(x)-2 c_1 \log (x)+4+c_1{}^2\right )}} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(x*(-x*y(x)**2 + 1))*y(x) - 1)*y(x)/(2*x) cannot be solved by the factorable group method