8.13.4 problem problem 7

Internal problem ID [925]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 7.2, Matrices and Linear systems. Page 417
Problem number : problem 7
Date solved : Monday, January 27, 2025 at 03:22:33 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 63

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 124

DSolve[{D[x[t],t]==y[t]+z[t],D[y[t],t]==z[t]+x[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}{3} e^{-t} \left (c_1 \left (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ z(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t}+2\right )\right ) \\ \end{align*}