70.4.19 problem 25

Internal problem ID [18700]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 25
Date solved : Thursday, October 02, 2025 at 03:21:55 PM
CAS classification : [[_homogeneous, `class D`], _rational]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 293

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

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

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 46

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2+1}{K[1] \left (K[1]^2+3\right )}dK[1]=-x-\log (x)-1+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out