85.33.17 problem 17

Internal problem ID [22640]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 17
Date solved : Thursday, October 02, 2025 at 08:57:16 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} y^{\prime }&=\frac {2 y x -y^{4}}{3 x^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 86

ode:=diff(y(x),x) = 1/3*(2*x*y(x)-y(x)^4)/x^2; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 71

ode=D[y[x],x]==(2*x*y[x]-y[x]^4)/(3*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^{2/3}}{\sqrt [3]{x+c_1}}\\ y(x)&\to -\frac {\sqrt [3]{-1} x^{2/3}}{\sqrt [3]{x+c_1}}\\ y(x)&\to \frac {(-1)^{2/3} x^{2/3}}{\sqrt [3]{x+c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.954 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (2*x*y(x) - y(x)**4)/(3*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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