52.10.8 problem 8

Internal problem ID [8402]
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 : 8
Date solved : Monday, January 27, 2025 at 03:57:34 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-7 y\\ y^{\prime }&=5 x \left (t \right )+10 y+4 z \left (t \right )\\ z^{\prime }\left (t \right )&=5 y+2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 65

dsolve([diff(x(t),t)=2*x(t)-7*y(t),diff(y(t),t)=5*x(t)+10*y(t)+4*z(t),diff(z(t),t)=5*y(t)+2*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {7 c_{2} {\mathrm e}^{7 t}}{5}-\frac {7 c_3 \,{\mathrm e}^{5 t}}{3}+c_{1} {\mathrm e}^{2 t} \\ y &= c_{2} {\mathrm e}^{7 t}+c_3 \,{\mathrm e}^{5 t} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{7 t}+\frac {5 c_3 \,{\mathrm e}^{5 t}}{3}-\frac {5 c_{1} {\mathrm e}^{2 t}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 190

DSolve[{D[x[t],t]==2*x[t]-7*y[t],D[y[t],t]==5*x[t]+10*y[t]+4*z[t],D[z[t],t]==5*y[t]+2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {1}{30} e^{2 t} \left (5 c_1 \left (-35 e^{3 t}+21 e^{5 t}+8\right )+7 \left (-5 (3 c_2+4 c_3) e^{3 t}+3 (5 c_2+4 c_3) e^{5 t}+8 c_3\right )\right ) \\ y(t)\to \frac {1}{2} e^{5 t} \left (5 c_1 \left (e^{2 t}-1\right )+c_2 \left (5 e^{2 t}-3\right )+4 c_3 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{6} e^{2 t} \left (5 c_1 \left (-5 e^{3 t}+3 e^{5 t}+2\right )-5 (3 c_2+4 c_3) e^{3 t}+3 (5 c_2+4 c_3) e^{5 t}+14 c_3\right ) \\ \end{align*}