52.10.13 problem 12

Internal problem ID [8407]
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 : 12
Date solved : Monday, January 27, 2025 at 03:57:39 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-x \left (t \right )+4 y+2 z \left (t \right )\\ y^{\prime }&=4 x \left (t \right )-y-2 z \left (t \right )\\ z^{\prime }\left (t \right )&=6 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 57

dsolve([diff(x(t),t)=-x(t)+4*y(t)+2*z(t),diff(y(t),t)=4*x(t)-y(t)-2*z(t),diff(z(t),t)=6*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} c_{2} +c_{1} {\mathrm e}^{-5 t}+\frac {2 c_3 \,{\mathrm e}^{6 t}}{11} \\ y &= -\frac {2 c_3 \,{\mathrm e}^{6 t}}{11}+{\mathrm e}^{3 t} c_{2} -c_{1} {\mathrm e}^{-5 t} \\ z \left (t \right ) &= c_3 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 105

DSolve[{D[x[t],t]==-x[t]+4*y[t]+2*z[t],D[y[t],t]==4*x[t]-y[t]-2*z[t],D[z[t],t]==6*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{22} e^{-5 t} \left (11 c_1 \left (e^{8 t}+1\right )+11 c_2 \left (e^{8 t}-1\right )+4 c_3 \left (e^{11 t}-1\right )\right ) \\ y(t)\to \frac {1}{22} e^{-5 t} \left (11 c_1 \left (e^{8 t}-1\right )+11 c_2 \left (e^{8 t}+1\right )-4 c_3 \left (e^{11 t}-1\right )\right ) \\ z(t)\to c_3 e^{6 t} \\ \end{align*}