35.8.7 problem 7

Internal problem ID [6214]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number : 7
Date solved : Sunday, March 30, 2025 at 10:43:20 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 85

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

\begin{align*} y &= \frac {3^{{2}/{3}} \left (3 c_1 \,x^{4}-x \right )^{{1}/{3}}}{3 x} \\ y &= -\frac {3^{{2}/{3}} \left (3 c_1 \,x^{4}-x \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{6 x} \\ y &= \frac {\left (3 i 3^{{1}/{6}}-3^{{2}/{3}}\right ) \left (3 c_1 \,x^{4}-x \right )^{{1}/{3}}}{6 x} \\ \end{align*}

✓ Mathematica. Time used: 0.789 (sec). Leaf size: 85

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

✓ Sympy. Time used: 1.711 (sec). Leaf size: 80

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

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