23.1.721 problem 717

Internal problem ID [5328]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 717
Date solved : Tuesday, September 30, 2025 at 12:30:19 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x \left (1-x^{2} y^{4}\right ) y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 133

ode:=x*(1-y(x)^4*x^2)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {2}\, \sqrt {\left (x -\sqrt {-4 c_1^{2}+x^{2}}\right ) c_1 x}}{2 c_1 x} \\ y &= \frac {\sqrt {2}\, \sqrt {\left (x -\sqrt {-4 c_1^{2}+x^{2}}\right ) c_1 x}}{2 c_1 x} \\ y &= -\frac {\sqrt {2}\, \sqrt {c_1 x \left (x +\sqrt {-4 c_1^{2}+x^{2}}\right )}}{2 c_1 x} \\ y &= \frac {\sqrt {2}\, \sqrt {c_1 x \left (x +\sqrt {-4 c_1^{2}+x^{2}}\right )}}{2 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 12.466 (sec). Leaf size: 144

ode=x(1-x^2 y[x]^4)D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {c_1-\frac {\sqrt {x^2 \left (-1+c_1{}^2 x^2\right )}}{x^2}}\\ y(x)&\to \sqrt {c_1-\frac {\sqrt {x^2 \left (-1+c_1{}^2 x^2\right )}}{x^2}}\\ y(x)&\to -\sqrt {\frac {\sqrt {x^2 \left (-1+c_1{}^2 x^2\right )}}{x^2}+c_1}\\ y(x)&\to \sqrt {\frac {\sqrt {x^2 \left (-1+c_1{}^2 x^2\right )}}{x^2}+c_1}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 5.010 (sec). Leaf size: 126

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(-x**2*y(x)**4 + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} - 4}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} - 4}}{x}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} - 4}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} - 4}}{x}}}{2}\right ] \]