35.4.3 problem 3

Internal problem ID [6121]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 4. OTHER METHODS FOR FIRST-ORDER EQUATIONS. page 406
Problem number : 3
Date solved : Sunday, March 30, 2025 at 10:39:26 AM
CAS classification : [_separable]

\begin{align*} 3 x y^{2} y^{\prime }+3 y^{3}&=1 \end{align*}

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

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

\begin{align*} y &= \frac {\left (9 x^{3}+27 c_1 \right )^{{1}/{3}}}{3 x} \\ y &= -\frac {\left (9 x^{3}+27 c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{6 x} \\ y &= \frac {\left (9 x^{3}+27 c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{6 x} \\ \end{align*}

✓ Mathematica. Time used: 0.299 (sec). Leaf size: 195

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

\begin{align*} y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{x^3+e^{9 c_1}}}{x} \\ y(x)\to \frac {\sqrt [3]{x^3+e^{9 c_1}}}{\sqrt [3]{3} x} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{x^3+e^{9 c_1}}}{\sqrt [3]{3} x} \\ y(x)\to -\sqrt [3]{-\frac {1}{3}} \\ y(x)\to \frac {1}{\sqrt [3]{3}} \\ y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{3}} \\ y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{x^3}}{x} \\ y(x)\to \frac {\sqrt [3]{x^3}}{\sqrt [3]{3} x} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{x^3}}{\sqrt [3]{3} x} \\ \end{align*}

✓ Sympy. Time used: 1.593 (sec). Leaf size: 75

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

\[ \left [ y{\left (x \right )} = \frac {3^{\frac {2}{3}} \sqrt [3]{\frac {C_{1}}{x^{3}} + 1}}{3}, \ y{\left (x \right )} = \frac {\left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 1}}{6}, \ y{\left (x \right )} = \frac {\left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} + 1}}{6}\right ] \]