14.27.4 problem 4
Internal
problem
ID
[2792]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.1
(Introduction).
Page
3770
Problem
number
:
4
Date
solved
:
Sunday, March 30, 2025 at 12:20:35 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-x \left (t \right ) y \left (t \right )^{2}\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right )-y \left (t \right ) x \left (t \right )^{2}\\ \frac {d}{d t}z \left (t \right )&=1-z \left (t \right )+x \left (t \right )^{2} \end{align*}
✓ Maple. Time used: 0.774 (sec). Leaf size: 228
ode:=[diff(x(t),t) = -x(t)-x(t)*y(t)^2, diff(y(t),t) = -y(t)-y(t)*x(t)^2, diff(z(t),t) = 1-z(t)+x(t)^2];
dsolve(ode);
\begin{align*}
\\
\left [\left \{x \left (t \right ) &= \frac {\sqrt {\left ({\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}-1\right ) c_2 \,{\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}}\, {\mathrm e}^{-t}}{{\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}-1}, x \left (t \right ) = -\frac {\sqrt {\left ({\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}-1\right ) c_2 \,{\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}}\, {\mathrm e}^{-t}}{{\mathrm e}^{-2 c_3 c_2} {\mathrm e}^{-{\mathrm e}^{-2 t} c_2}-1}\right \}, \left \{y \left (t \right ) = \frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )+x \left (t \right )\right )}}{x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {-x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )+x \left (t \right )\right )}}{x \left (t \right )}\right \}, \{z \left (t \right ) = \left (\int {\mathrm e}^{t} \left (x \left (t \right )^{2}+1\right )d t +c_1 \right ) {\mathrm e}^{-t}\}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.211 (sec). Leaf size: 445
ode={D[x[t],t]==-x[t]-x[t]*y[t]^2,D[y[t],t]==-y[t]-y[t]*x[t]^2,D[z[t],t]==1-z[t]+x[t]^2};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to -\sqrt {W\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (e^{K[1]^2+2 c_1} K[1]^2\right )+1\right )}dK[1]\&\right ][-t+c_2]{}^2 \exp \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (e^{K[1]^2+2 c_1} K[1]^2\right )+1\right )}dK[1]\&\right ][-t+c_2]{}^2+2 c_1\right )\right )} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (e^{K[1]^2+2 c_1} K[1]^2\right )+1\right )}dK[1]\&\right ][-t+c_2] \\
z(t)\to e^{-t} \left (\int _1^te^{K[3]} \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (e^{K[1]^2+2 c_1} K[1]^2\right )+1\right )}dK[1]\&\right ][c_2-K[3]]{}^2+1\right )dK[3]+c_3\right ) \\
y(t)\to \sqrt {W\left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \left (W\left (e^{K[2]^2+2 c_1} K[2]^2\right )+1\right )}dK[2]\&\right ][-t+c_2]{}^2 \exp \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \left (W\left (e^{K[2]^2+2 c_1} K[2]^2\right )+1\right )}dK[2]\&\right ][-t+c_2]{}^2+2 c_1\right )\right )} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \left (W\left (e^{K[2]^2+2 c_1} K[2]^2\right )+1\right )}dK[2]\&\right ][-t+c_2] \\
z(t)\to e^{-t} \left (\int _1^te^{K[4]} \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[2] \left (W\left (e^{K[2]^2+2 c_1} K[2]^2\right )+1\right )}dK[2]\&\right ][c_2-K[4]]{}^2+1\right )dK[4]+c_3\right ) \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(x(t)*y(t)**2 + x(t) + Derivative(x(t), t),0),Eq(x(t)**2*y(t) + y(t) + Derivative(y(t), t),0),Eq(-x(t)**2 + z(t) + Derivative(z(t), t) - 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
NotImplementedError :