52.10.27 problem 28

Internal problem ID [8421]
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 : 28
Date solved : Monday, January 27, 2025 at 04:00:18 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 57

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

\begin{align*} x(t)\to \frac {1}{2} e^{4 t} (t (c_3 t+2 c_2)+2 c_1) \\ y(t)\to e^{4 t} (c_3 t+c_2) \\ z(t)\to c_3 e^{4 t} \\ \end{align*}