52.10.26 problem 27

Internal problem ID [8420]
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 : 27
Date solved : Monday, January 27, 2025 at 04:00:17 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 65

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

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