64.19.2 problem 2

Internal problem ID [13643]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.7. Exercises page 375
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 05:54:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-3 x \left (t \right )-6 y \left (t \right )+6 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 73

dsolve([diff(x(t),t)=x(t)-y(t)-z(t),diff(y(t),t)=x(t)+3*y(t)+z(t),diff(z(t),t)=-3*x(t)-6*y(t)+6*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} c_{1} +c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -{\mathrm e}^{5 t} c_{1} -\frac {7 c_{2} {\mathrm e}^{2 t}}{10}-c_{3} {\mathrm e}^{3 t} \\ z &= -3 \,{\mathrm e}^{5 t} c_{1} -\frac {3 c_{2} {\mathrm e}^{2 t}}{10}-c_{3} {\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 217

DSolve[{D[x[t],t]==x[t]-y[t]-z[t],D[y[t],t]==x[t]+3*y[t]+z[t],D[z[t],t]==3*x[t]-6*y[t]+6*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-50 (c_1+c_2)\right ) \\ y(t)\to \frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-5 (c_1+c_2)\right ) \\ z(t)\to (c_1+c_2) \left (-e^{2 t}\right )+(c_1+c_2+c_3) e^{4 t} \cos \left (\sqrt {5} t\right )+\frac {(c_1-8 c_2+2 c_3) e^{4 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} \\ \end{align*}