14.30.6 problem 6
Internal
problem
ID
[2819]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.3
(Stability
of
equilibrium
solutions).
Page
393
Problem
number
:
6
Date
solved
:
Tuesday, January 28, 2025 at 02:38:54 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&={\mathrm e}^{y \left (t \right )}-x\\ y^{\prime }\left (t \right )&={\mathrm e}^{x}+y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.547 (sec). Leaf size: 63
dsolve([diff(x(t),t)=exp(y(t))-x(t),diff(y(t),t)=exp(x(t))+y(t)],singsol=all)
\begin{align*}
\left \{y &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\frac {-\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_1}{\textit {\_f}}} {\mathrm e}^{\frac {{\mathrm e}^{\textit {\_f}}}{\textit {\_f}}}}{\textit {\_f}}\right ) \textit {\_f} +{\mathrm e}^{\textit {\_f}}+c_1}{\textit {\_f}}}+\textit {\_f}}d \textit {\_f} +t +c_2 \right )\right \} \\
\{x \left (t \right ) &= \ln \left (y^{\prime }-y\right )\} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.515 (sec). Leaf size: 860
DSolve[{D[x[t],t]==Exp[y[t]]-x[t],D[y[t],t]==Exp[x[t]]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
y(t)\to -\frac {\exp \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]\right )+W\left (-\frac {\exp \left (-\frac {\exp \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]\right )+c_1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]}\right )}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]}\right ) \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]+c_1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2]} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right )}{\exp \left (\frac {c_1}{K[1]}+W\left (-\frac {e^{-\frac {c_1}{K[1]}-\frac {e^{K[1]}}{K[1]}}}{K[1]}\right )+\frac {e^{K[1]}}{K[1]}\right ) K[1]-1}dK[1]\&\right ][-t+c_2] \\
\end{align*}