52.10.12 problem 11

Internal problem ID [8406]
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 : 11
Date solved : Monday, January 27, 2025 at 03:57:38 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-x \left (t \right )-y\\ y^{\prime }&=\frac {3 x \left (t \right )}{4}-\frac {3 y}{2}+3 z \left (t \right )\\ z^{\prime }\left (t \right )&=\frac {x \left (t \right )}{8}+\frac {y}{4}-\frac {z \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 66

dsolve([diff(x(t),t)=-x(t)-y(t),diff(y(t),t)=3/4*x(t)-3/2*y(t)+3*z(t),diff(z(t),t)=1/8*x(t)+1/4*y(t)-1/2*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-\frac {3 t}{2}} \\ y &= -\frac {c_{1} {\mathrm e}^{-\frac {t}{2}}}{2}+\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{2}}}{2} \\ z \left (t \right ) &= -\frac {5 c_{1} {\mathrm e}^{-\frac {t}{2}}}{12}-\frac {c_{2} {\mathrm e}^{-t}}{4}-\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{2}}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 168

DSolve[{D[x[t],t]==-x[t]-y[t],D[y[t],t]==3/4*x[t]-3/2*y[t]+3*z[t],D[z[t],t]==1/8x[t]+1/4*y[t]-1/2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

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