76.25.3 problem 3

Internal problem ID [17816]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:03:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )+2 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.149 (sec). Leaf size: 66

dsolve([diff(x__1(t),t)=2*x__1(t)-4*x__2(t)+2*x__3(t),diff(x__2(t),t)=-4*x__1(t)+2*x__2(t)-2*x__3(t),diff(x__3(t),t)=2*x__1(t)-2*x__2(t)-x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{7 t}+c_{3} {\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= -c_{2} {\mathrm e}^{7 t}-c_{3} {\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{7 t}}{2}-4 c_{3} {\mathrm e}^{-2 t}+2 c_{1} {\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 137

DSolve[{D[x1[t],t]==2*x1[t]-4*x2[t]+2*x3[t],D[x2[t],t]==-4*x1[t]+2*x2[t]-2*x3[t],D[x3[t],t]==2*x1[t]-2*x2[t]-x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (4 e^{9 t}+5\right )-2 (2 c_2-c_3) \left (e^{9 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-2 t} \left (-4 c_1 \left (e^{9 t}-1\right )+c_2 \left (4 e^{9 t}+5\right )-2 c_3 \left (e^{9 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-2 t} \left (2 c_1 \left (e^{9 t}-1\right )-2 c_2 \left (e^{9 t}-1\right )+c_3 \left (e^{9 t}+8\right )\right ) \\ \end{align*}