75.29.6 problem 807

Internal problem ID [17260]
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
Date solved : Tuesday, January 28, 2025 at 09:58:32 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.156 (sec). Leaf size: 57

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 ) &= {\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= {\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{t}+c_{1} {\mathrm e}^{-2 t} \\ z &= -2 \,{\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{t}-c_{1} {\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 124

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