12.6.4 problem 4
Internal
problem
ID
[1683]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Section
2.5
Page
79
Problem
number
:
4
Date
solved
:
Saturday, March 29, 2025 at 11:29:52 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 x -2 y^{2}+\left (12 y^{2}-4 x y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 426
ode:=2*x-2*y(x)^2+(12*y(x)^2-4*x*y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-6 x^{5}-6 c_1 \,x^{3}+81 x^{4}+162 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}{6}+\frac {x^{2}}{6 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-6 x^{5}-6 c_1 \,x^{3}+81 x^{4}+162 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}+\frac {x}{6} \\
y &= \frac {\left (x -\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}\right ) \left (i \left (\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}+x \right ) \sqrt {3}-x +\left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}\right )}{12 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}}{12}-\frac {x \left (i x \sqrt {3}+x -2 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}\right )}{12 \left (-27 x^{2}-27 c_1 +x^{3}+3 \sqrt {-3 \left (x^{2}+c_1 \right ) \left (2 x^{3}-27 x^{2}-27 c_1 \right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.898 (sec). Leaf size: 414
ode=(2*x-2*y[x]^2)+(12*y[x]^2-4*x*y[x])*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {x^2}{3\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}-\frac {\sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {x}{6} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x + (-4*x*y(x) + 12*y(x)**2)*Derivative(y(x), x) - 2*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out