7.23.14 problem 20

Internal problem ID [600]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 20
Date solved : Monday, January 27, 2025 at 02:55:09 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 90

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} c_3 -2 c_2 \,{\mathrm e}^{-t}-\frac {{\mathrm e}^{-t}}{3}+\frac {2 t \,{\mathrm e}^{-t}}{3}-{\mathrm e}^{-t} c_1 \\ y \left (t \right ) &= {\mathrm e}^{2 t} c_3 +c_2 \,{\mathrm e}^{-t}-\frac {t \,{\mathrm e}^{-t}}{3} \\ z \left (t \right ) &= {\mathrm e}^{2 t} c_3 +c_2 \,{\mathrm e}^{-t}-\frac {t \,{\mathrm e}^{-t}}{3}+{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 164

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

\begin{align*} x(t)\to \frac {1}{9} e^{-t} \left (6 t+3 c_1 \left (e^{3 t}+2\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1-3 c_2-3 c_3\right ) \\ y(t)\to \frac {1}{9} e^{-t} \left (-3 t+3 c_1 \left (e^{3 t}-1\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1+6 c_2-3 c_3\right ) \\ z(t)\to \frac {1}{9} e^{-t} \left (-3 t+3 c_1 \left (e^{3 t}-1\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1-3 c_2+6 c_3\right ) \\ \end{align*}