Internal problem ID [11966]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+3 y \left (t \right )-4 z \left (t \right )\\ z^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right )-4 z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 65
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 ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+2 c_{2} {\mathrm e}^{2 t}+7 c_{3} {\mathrm e}^{-3 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+11 c_{3} {\mathrm e}^{-3 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 198
DSolve[{x'[t]==x[t]+y[t]-z[t],y'[t]==2*x[t]+3*y[t]-4*z[t],z'[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*}