Internal problem ID [15530]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of differential equations). Section 22. Integration of homogeneous
linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number: 807.
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 )&=x \left (t \right )-y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 58
dsolve([diff(x(t),t)=-x(t)+y(t)+z(t),diff(y(t),t)=x(t)-y(t)+z(t),diff(z(t),t)=x(t)+y(t)-z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{t}-2 c_{3} {\mathrm e}^{-2 t}-c_{1} {\mathrm e}^{-2 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 124
DSolve[{x'[t]==-x[t]+y[t]+z[t],y'[t]==x[t]-y[t]+z[t],z'[t]==x[t]+y[t]-z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ z(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t}+2\right )\right ) \\ \end{align*}