Internal problem ID [15512]
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 20. The method of elimination.
Exercises page 212
Problem number: 781.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 56
dsolve([diff(x(t),t)=-y(t)+z(t),diff(y(t),t)=z(t),diff(z(t),t)=-x(t)+z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{3} \sin \left (t \right )-c_{2} \cos \left (t \right )+c_{3} \cos \left (t \right ) \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} \cos \left (t \right )+c_{3} \sin \left (t \right ) \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} \sin \left (t \right )+c_{3} \cos \left (t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 112
DSolve[{x'[t]==-y[t]+z[t],y'[t]==z[t],z'[t]==-x[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_1 \cos (t)+(c_3-c_2) \sin (t) \\ y(t)\to \frac {1}{2} \left ((-c_1+c_2+c_3) e^t+(c_1+c_2-c_3) \cos (t)+(c_1-c_2+c_3) \sin (t)\right ) \\ z(t)\to \frac {1}{2} \left ((-c_1+c_2+c_3) e^t+(c_1-c_2+c_3) \cos (t)-(c_1+c_2-c_3) \sin (t)\right ) \\ \end{align*}