41.2.56 problem 52

Internal problem ID [8758]
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 : 52
Date solved : Tuesday, September 30, 2025 at 05:50:19 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.064 (sec). Leaf size: 97

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

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_1^{2}-4 x^{2}}+2 c_1}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_1^{2}-4 x^{2}}+2 c_1}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_1^{2}-4 x^{2}}+2 c_1}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_1^{2}-4 x^{2}}+2 c_1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.005 (sec). Leaf size: 122

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

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

✓ Sympy. Time used: 2.745 (sec). Leaf size: 75

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

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