81.2.7 problem 3-8
Internal
problem
ID
[21496]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
3.
Exact
differential
equations.
Page
42.
Problem
number
:
3-8
Date
solved
:
Thursday, October 02, 2025 at 07:42:08 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 3 x^{2}+6 x y^{2}+\left (6 x^{2} y+4 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 505
ode:=3*x^2+6*x*y(x)^2+(6*x^2*y(x)+4*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9 x^{9}+9 c_1 \,x^{6}+4 x^{6}+8 c_1 \,x^{3}+4 c_1^{2}}\right )^{{1}/{3}}}{4}+\frac {9 x^{4}}{4 \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9 x^{9}+9 c_1 \,x^{6}+4 x^{6}+8 c_1 \,x^{3}+4 c_1^{2}}\right )^{{1}/{3}}}-\frac {3 x^{2}}{4} \\
y &= \frac {9 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{2}/{3}}-9 x^{4}-6 x^{2} \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{1}/{3}}-\left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{2}/{3}}}{8 \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{1}/{3}}} \\
y &= -\frac {9 i \sqrt {3}\, x^{4}+9 x^{4}-i \sqrt {3}\, \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{2}/{3}}+6 x^{2} \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{1}/{3}}+\left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{2}/{3}}}{8 \left (-24 x^{3}-24 c_1 -27 x^{6}+12 \sqrt {9}\, \sqrt {\left (x^{6}+\frac {4}{9} x^{3}+\frac {4}{9} c_1 \right ) \left (x^{3}+c_1 \right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.645 (sec). Leaf size: 459
ode=3*x^2+6*x*y[x]^2+(6*x^2*y[x]+4*y[x]^2)*D[y[x],x] ==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{4} \left (-3 x^2+\sqrt [3]{3} \sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}+\frac {3\ 3^{2/3} x^4}{\sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}}\right )\\ y(x)&\to \frac {1}{16} \left (-12 x^2+2 i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}-\frac {6 \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^4}{\sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}}\right )\\ y(x)&\to \frac {1}{16} \left (-12 x^2-2 \sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}+\frac {6 i 3^{2/3} \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-9 x^6-8 x^3+4 \sqrt {9 x^9+(4-9 c_1) x^6-8 c_1 x^3+4 c_1{}^2}+8 c_1}}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2 + 6*x*y(x)**2 + (6*x**2*y(x) + 4*y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out