50.29.9 problem 3(e)

Internal problem ID [8205]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 3(e)
Date solved : Monday, January 27, 2025 at 03:45:48 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )-y \left (t \right )+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )+z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 62

dsolve([diff(x(t),t)=3*x(t)+2*y(t)+z(t),diff(y(t),t)=-2*x(t)-y(t)+3*z(t),diff(z(t),t)=x(t)+y(t)+z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t} \\ y &= -\frac {c_{1} {\mathrm e}^{3 t}}{5}-\frac {7 c_{2} {\mathrm e}^{-t}}{3}-c_3 \,{\mathrm e}^{t} \\ z \left (t \right ) &= \frac {2 c_{1} {\mathrm e}^{3 t}}{5}+\frac {2 c_{2} {\mathrm e}^{-t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 180

DSolve[{D[x[t],t]==3*x[t]+2*y[t]+z[t],D[y[t],t]==-2*x[t]-y[t]+3*z[t],D[z[t],t]==x[t]+y[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{8} e^{-t} \left (c_1 \left (6 e^{2 t}+5 e^{4 t}-3\right )+\left (e^{2 t}-1\right ) \left (c_2 \left (5 e^{2 t}+3\right )+2 c_3 \left (5 e^{2 t}-3\right )\right )\right ) \\ y(t)\to \frac {1}{8} e^{-t} \left (-\left (c_1 \left (6 e^{2 t}+e^{4 t}-7\right )\right )+c_2 \left (2 e^{2 t}-e^{4 t}+7\right )-2 c_3 \left (-8 e^{2 t}+e^{4 t}+7\right )\right ) \\ z(t)\to \frac {1}{4} e^{-t} \left (c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}-1\right )+2 c_3 \left (e^{4 t}+1\right )\right ) \\ \end{align*}