52.10.29 problem 30

Internal problem ID [8423]
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 : 30
Date solved : Monday, January 27, 2025 at 04:00:20 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 2\\ z \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 36

dsolve([diff(x(t),t) = z(t), diff(y(t),t) = y(t), diff(z(t),t) = x(t), x(0) = 1, y(0) = 2, z(0) = 5], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 42

DSolve[{D[x[t],t]==z[t],D[y[t],t]==y[t],D[z[t],t]==x[t]},{x[0]==1,y[0]==2,z[0]==5},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 3 e^t-2 e^{-t} \\ z(t)\to 2 e^{-t}+3 e^t \\ y(t)\to 2 e^t \\ \end{align*}