38.2.8 problem 8

Internal problem ID [6437]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 12:43:22 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 90

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

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

✓ Sympy. Time used: 1.460 (sec). Leaf size: 63

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)
 

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