52.9.16 problem 16

Internal problem ID [8394]
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 8.1. Page 332
Problem number : 16
Date solved : Monday, January 27, 2025 at 03:57:28 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.101 (sec). Leaf size: 64

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 76

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

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