84.10.8 problem 5.18

Internal problem ID [22136]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 5. Homogeneous differential equations. Supplementary problems
Problem number : 5.18
Date solved : Friday, October 03, 2025 at 07:59:48 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 74

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

\[ -\frac {\ln \left (27 x^{2} y+8\right )}{6}-\frac {\ln \left (3 \left (x y^{2}\right )^{{2}/{3}}+2 y\right )}{3}+\frac {\ln \left (y \left (9 \left (x y^{2}\right )^{{1}/{3}} y x -6 \left (x y^{2}\right )^{{2}/{3}}+4 y\right )\right )}{6}+\frac {\ln \left (y\right )}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 29.913 (sec). Leaf size: 382

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

\begin{align*} y(x)&\to \frac {3 x^2}{\sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}}+3 e^{-6 c_1} \sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}+2 e^{-2 c_1}\\ y(x)&\to \frac {1}{2} \left (-\frac {3 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}}+3 i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}+4 e^{-2 c_1}\right )\\ y(x)&\to \frac {1}{2} \left (\frac {3 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}}-3 \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{10 c_1} x^2+\sqrt {e^{18 c_1} x^4 \left (-x^2+e^{2 c_1}\right )}}+4 e^{-2 c_1}\right )\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)**2/(x*y(x) + (x*y(x)**2)**(1/3)) cann