62.3.4 problem Ex 4

Internal problem ID [12739]
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 10. Homogeneous equations. Page 15
Problem number : Ex 4
Date solved : Monday, March 31, 2025 at 06:58:36 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 2 x^{2} y+y^{3}-x^{3} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

ode:=2*x^2*y(x)+y(x)^3-x^3*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 47

ode=2*x^2*y[x]+y[x]^3-x^3*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.665 (sec). Leaf size: 32

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

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