52.10.7 problem 7

Internal problem ID [8401]
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 : 7
Date solved : Monday, January 27, 2025 at 03:57:33 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= \frac {2 c_3 \,{\mathrm e}^{2 t}}{3}+c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{-t}}{2} \\ y &= c_3 \,{\mathrm e}^{2 t} \\ z \left (t \right ) &= \frac {c_3 \,{\mathrm e}^{2 t}}{3}+c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 88

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

\begin{align*} x(t)\to \frac {1}{6} e^{-t} \left (4 c_2 e^{3 t}+(6 c_1-3 (c_2+c_3)) e^{2 t}-c_2+3 c_3\right ) \\ y(t)\to c_2 e^{2 t} \\ z(t)\to \frac {1}{3} e^{-t} \left (c_2 \left (e^{3 t}-1\right )+3 c_3\right ) \\ \end{align*}