75.27.7 problem 782

Internal problem ID [17243]
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
Date solved : Tuesday, January 28, 2025 at 09:58:24 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.137 (sec). Leaf size: 63

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 124

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