40.2.19 problem 44

Internal problem ID [6597]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary problems. Page 22
Problem number : 44
Date solved : Sunday, March 30, 2025 at 11:11:21 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 90

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

\begin{align*} y &= \frac {2^{{1}/{3}} {\left (-\left (x^{4}-4 c_1 \right ) x^{2}\right )}^{{1}/{3}}}{2 x} \\ y &= -\frac {2^{{1}/{3}} {\left (-\left (x^{4}-4 c_1 \right ) x^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y &= \frac {2^{{1}/{3}} {\left (-\left (x^{4}-4 c_1 \right ) x^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 99

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

\begin{align*} y(x)\to \frac {\sqrt [3]{-x^4+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-x^4+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-x^4+4 c_1}}{2^{2/3} \sqrt [3]{x}} \\ \end{align*}

✓ Sympy. Time used: 1.466 (sec). Leaf size: 75

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

\[ \left [ y{\left (x \right )} = \frac {\sqrt [3]{2} \sqrt [3]{\frac {C_{1}}{x} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x} - x^{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x} - x^{3}}}{4}\right ] \]