23.1.682 problem 677

Internal problem ID [5289]
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 : 677
Date solved : Tuesday, September 30, 2025 at 12:07:27 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (1-x^{4} y^{2}\right ) y^{\prime }&=x^{3} y^{3} \end{align*}

✓ Maple. Time used: 0.187 (sec). Leaf size: 157

ode:=(1-x^4*y(x)^2)*diff(y(x),x) = x^3*y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 11.667 (sec). Leaf size: 122

ode=(1-x^4 y[x]^2)D[y[x],x]==x^3 y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {\frac {1-\sqrt {1+4 c_1 x^4}}{x^4}}\\ y(x)&\to \sqrt {\frac {1-\sqrt {1+4 c_1 x^4}}{x^4}}\\ y(x)&\to -\sqrt {\frac {1+\sqrt {1+4 c_1 x^4}}{x^4}}\\ y(x)&\to \sqrt {\frac {1+\sqrt {1+4 c_1 x^4}}{x^4}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 2.791 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*y(x)**3 + (-x**4*y(x)**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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