80.3.34 problem 37

Internal problem ID [21198]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 37
Date solved : Thursday, October 02, 2025 at 07:26:15 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} y^{2}+\left (y x +3 y^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 228

ode:=y(x)^2+(x*y(x)+3*y(x)^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{2}/{3}}-12 x}{6 \left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{1}/{3}}} \\ y &= -\frac {i \sqrt {3}\, \left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}\, x +\left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{2}/{3}}-12 x}{12 \left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}\, \left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}\, x -\left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{2}/{3}}+12 x}{12 \left (108 c_1 +12 \sqrt {12 x^{3}+81 c_1^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 1.191 (sec). Leaf size: 293

ode=(y[x]^2)+(x*y[x]+3*y[x]^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0\\ y(x)&\to \frac {-2 \sqrt [3]{3} x+\sqrt [3]{2} \left (\sqrt {12 x^3+81 c_1{}^2}+9 c_1\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {12 x^3+81 c_1{}^2}+9 c_1}}\\ y(x)&\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {12 x^3+81 c_1{}^2}+9 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x}{12 \sqrt [3]{\sqrt {12 x^3+81 c_1{}^2}+9 c_1}}\\ y(x)&\to \frac {2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {12 x^3+81 c_1{}^2}+9 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x}{12 \sqrt [3]{\sqrt {12 x^3+81 c_1{}^2}+9 c_1}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out