Internal problem ID [15531]
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: 808.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )-z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 52
dsolve([diff(x(t),t)=2*x(t)-y(t)+z(t),diff(y(t),t)=x(t)+2*y(t)-z(t),diff(z(t),t)=x(t)-y(t)+2*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{3} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{t} \\ z \left (t \right ) &= c_{3} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 99
DSolve[{x'[t]==2*x[t]-y[t]+z[t],y'[t]==x[t]+2*y[t]-z[t],z'[t]==x[t]-y[t]+2*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{2 t} \left (c_1-(c_2-c_3) \left (e^t-1\right )\right ) \\ y(t)\to e^t \left (c_1 \left (e^t-1\right )+(c_2-c_3) e^t+c_3\right ) \\ z(t)\to e^t \left (c_1 \left (e^t-1\right )+(c_2-c_3) e^t+(c_3-c_2) e^{2 t}+c_3\right ) \\ \end{align*}