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 : Friday, March 14, 2025 at 01:26:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&={\mathrm e}^{y \left (t \right )}-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&={\mathrm e}^{x \left (t \right )}+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.547 (sec). Leaf size: 63

ode:=[diff(x(t),t) = exp(y(t))-x(t), diff(y(t),t) = exp(x(t))+y(t)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.515 (sec). Leaf size: 860

ode={D[x[t],t]==Exp[y[t]]-x[t],D[y[t],t]==Exp[x[t]]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - exp(y(t)) + Derivative(x(t), t),0),Eq(-y(t) - exp(x(t)) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError : The given ODE Derivative(v(u), u) - (-v(u) - exp(u))/(u - exp(v(u))) cannot be solved by the factorable group method