12.5.37 problem 34
Internal
problem
ID
[1661]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Transformation
of
Nonlinear
Equations
into
Separable
Equations.
Section
2.4
Page
68
Problem
number
:
34
Date
solved
:
Saturday, March 29, 2025 at 11:18:32 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=\frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 272
ode:=diff(y(x),x) = (x^3+x^2*y(x)+3*y(x)^3)/(x^3+3*x*y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}-12\right ) x}{6 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (i \sqrt {3}\, \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}-12\right ) x}{12 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (\left (i \sqrt {3}-1\right ) \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}+12\right ) x}{12 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 45.333 (sec). Leaf size: 398
ode=D[y[x],x]==(x^3+x^2*y[x]+3*y[x]^3)/(x^3+3*x*y[x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {-2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\
y(x)\to \frac {2 \sqrt [3]{3} \left (1+i \sqrt {3}\right ) x^2+i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\
y(x)\to \frac {2 \sqrt [3]{3} \left (1-i \sqrt {3}\right ) x^2-\sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (x**3 + x**2*y(x) + 3*y(x)**3)/(x**3 + 3*x*y(x)**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : cannot determine truth value of Relational