28.4.12 problem 7.12

Internal problem ID [4544]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.12
Date solved : Sunday, March 30, 2025 at 03:25:54 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-y \left (t \right )&=0\\ -x \left (t \right )+\frac {d}{d t}y \left (t \right )&={\mathrm e}^{t}+{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.216 (sec). Leaf size: 37

ode:=[diff(x(t),t)-y(t) = 0, -x(t)+diff(y(t),t) = exp(t)+exp(-t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \sinh \left (t \right ) c_2 +\cosh \left (t \right ) c_1 -\cosh \left (t \right )+\sinh \left (t \right ) t \\ y \left (t \right ) &= \cosh \left (t \right ) c_2 +\sinh \left (t \right ) c_1 +\cosh \left (t \right ) t \\ \end{align*}

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 106

ode={D[x[t],t]-y[t]==0,-x[t]+D[y[t],t]==Exp[t]+Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (\left (e^{2 t}-1\right ) \log \left (e^{2 t}\right )+(-1+2 c_1+2 c_2) e^{2 t}-1+2 c_1-2 c_2\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (\left (e^{2 t}+1\right ) \log \left (e^{2 t}\right )+(1+2 c_1+2 c_2) e^{2 t}-1-2 c_1+2 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.171 (sec). Leaf size: 65

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

\[ \left [ x{\left (t \right )} = \frac {t e^{t}}{2} - \frac {t e^{- t}}{2} - \left (C_{1} + \frac {1}{4}\right ) e^{- t} + \left (C_{2} - \frac {1}{4}\right ) e^{t}, \ y{\left (t \right )} = \frac {t e^{t}}{2} + \frac {t e^{- t}}{2} + \left (C_{1} - \frac {1}{4}\right ) e^{- t} + \left (C_{2} + \frac {1}{4}\right ) e^{t}\right ] \]