Internal problem ID [15513]
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: 782.
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 )&=x \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 64
dsolve([diff(x(t),t)=y(t)+z(t),diff(y(t),t)=x(t)+z(t),diff(z(t),t)=x(t)+y(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t}+{\mathrm e}^{-t} c_{1} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{2 t}-2 c_{3} {\mathrm e}^{-t}-{\mathrm e}^{-t} c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 124
DSolve[{x'[t]==y[t]+z[t],y'[t]==x[t]+z[t],z'[t]==x[t]+y[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{3} e^{-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^{-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^{-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*}