52.10.10 problem 10

Internal problem ID [8404]
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 : 10
Date solved : Monday, January 27, 2025 at 03:57:36 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 128

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

\begin{align*} x(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ y(t)\to c_3 e^t \\ x(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ y(t)\to 0 \\ \end{align*}