52.10.9 problem 9

Internal problem ID [8403]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 9
Date solved : Monday, January 27, 2025 at 03:57:35 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 66

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

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{3 t} c_{2}}{4}+{\mathrm e}^{-t} c_{1} -c_3 \,{\mathrm e}^{-2 t} \\ y &= {\mathrm e}^{3 t} c_{2} +c_3 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= \frac {3 \,{\mathrm e}^{3 t} c_{2}}{4}-{\mathrm e}^{-t} c_{1} -3 c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 158

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

\begin{align*} x(t)\to \frac {1}{20} e^{-2 t} \left (c_1 \left (15 e^t+e^{5 t}+4\right )+4 c_2 \left (e^{5 t}-1\right )+c_3 \left (-5 e^t+e^{5 t}+4\right )\right ) \\ y(t)\to \frac {1}{5} e^{-2 t} \left (c_1 \left (e^{5 t}-1\right )+c_2 \left (4 e^{5 t}+1\right )+c_3 \left (e^{5 t}-1\right )\right ) \\ z(t)\to \frac {1}{20} e^{-2 t} \left (3 c_1 \left (-5 e^t+e^{5 t}+4\right )+12 c_2 \left (e^{5 t}-1\right )+c_3 \left (5 e^t+3 e^{5 t}+12\right )\right ) \\ \end{align*}