52.10.15 problem 14

Internal problem ID [8409]
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 : 14
Date solved : Monday, January 27, 2025 at 03:57:40 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 3\\ z \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 52

dsolve([diff(x(t),t) = x(t)+y(t)+4*z(t), diff(y(t),t) = 2*y(t), diff(z(t),t) = x(t)+y(t)+z(t), x(0) = 1, y(0) = 3, z(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 63

DSolve[{D[x[t],t]==x[t]+y[t]+4*z[t],D[y[t],t]==2*y[t],D[z[t],t]==x[t]+y[t]+z[t]},{x[0]==1,y[0]==3,z[0]==0},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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