75.29.7 problem 808

Internal problem ID [17261]
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
Date solved : Tuesday, January 28, 2025 at 09:58:33 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.165 (sec). Leaf size: 51

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 &= c_{3} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 99

DSolve[{D[x[t],t]==2*x[t]-y[t]+z[t],D[y[t],t]==x[t]+2*y[t]-z[t],D[z[t],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*}