64.19.1 problem 1

Internal problem ID [13642]
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 : 1
Date solved : Tuesday, January 28, 2025 at 05:54:14 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 )&=2 x \left (t \right )+3 y \left (t \right )-4 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=4 x \left (t \right )+y \left (t \right )-4 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 64

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= 7 \,{\mathrm e}^{-3 t} c_{1} +2 c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{t} \\ z &= 11 \,{\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 198

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

\begin{align*} x(t)\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-4 e^{5 t}-1\right )+2 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+c_3\right ) \\ y(t)\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-8 e^{5 t}-7\right )+4 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+7 c_3\right ) \\ z(t)\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-4 e^{5 t}-11\right )+2 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+11 c_3\right ) \\ \end{align*}