62.10.4 problem Ex 4

Internal problem ID [12762]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 17. Other forms which Integrating factors can be found. Page 25
Problem number : Ex 4
Date solved : Wednesday, March 05, 2025 at 08:26:28 PM
CAS classification : [_separable]

\begin{align*} x^{3} y-y^{4}+\left (y^{3} x -x^{4}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=x^3*y(x)-y(x)^4+(x*y(x)^3-x^4)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\left (1+i \sqrt {3}\right ) x}{2} \\ y &= \frac {\left (i \sqrt {3}-1\right ) x}{2} \\ y &= x \\ y &= c_{1} x \\ \end{align*}

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 99

ode=(x^3*y[x]-y[x]^4)+(y[x]^3*x-x^4)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to x \\ y(x)\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) x \\ y(x)\to \frac {1}{2} i \left (\sqrt {3}+i\right ) x \\ y(x)\to c_1 x \\ y(x)\to x \\ y(x)\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) x \\ y(x)\to \frac {1}{2} i \left (\sqrt {3}+i\right ) x \\ \end{align*}

✓ Sympy. Time used: 0.371 (sec). Leaf size: 5

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

\[ y{\left (x \right )} = C_{1} x \]