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 : Monday, January 27, 2025 at 09:23:17 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.489 (sec). Leaf size: 41

dsolve([diff(x(t),t)-y(t)=0,-x(t)+diff(y(t),t)=exp(t)+exp(-t)],singsol=all)
 

\begin{align*} x &= \sinh \left (t \right ) c_{2} +\cosh \left (t \right ) c_{1} +\sinh \left (t \right ) t -\frac {\cosh \left (t \right )}{2} \\ y &= \cosh \left (t \right ) c_{2} +\sinh \left (t \right ) c_{1} +t \cosh \left (t \right )+\frac {\sinh \left (t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.060 (sec). Leaf size: 106

DSolve[{D[x[t],t]-y[t]==0,-x[t]+D[y[t],t]==Exp[t]+Exp[-t]},{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*}