76.25.1 problem 1

Internal problem ID [17814]
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 : 1
Date solved : Tuesday, January 28, 2025 at 11:03:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.239 (sec). Leaf size: 64

dsolve([diff(x__1(t),t)=-4*x__1(t)+x__2(t)+0*x__3(t),diff(x__2(t),t)=1*x__1(t)-5*x__2(t)+x__3(t),diff(x__3(t),t)=0*x__1(t)+x__2(t)-4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +{\mathrm e}^{-4 t} c_{1} -\frac {c_{3} {\mathrm e}^{-6 t}}{2} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +c_{3} {\mathrm e}^{-6 t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} -{\mathrm e}^{-4 t} c_{1} -\frac {c_{3} {\mathrm e}^{-6 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 165

DSolve[{D[x1[t],t]==-4*x1[t]+x2[t]+0*x3[t],D[x2[t],t]==1*x1[t]-5*x2[t]+1*x3[t],D[x3[t],t]==0*x1[t]+x2[t]-4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{-6 t} \left (c_3 \left (2 e^t+1\right ) \left (e^t-1\right )^2+c_1 \left (3 e^{2 t}+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-6 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 ) \\ \text {x3}(t)\to \frac {1}{6} e^{-6 t} \left (c_1 \left (2 e^t+1\right ) \left (e^t-1\right )^2+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (3 e^{2 t}+2 e^{3 t}+1\right )\right ) \\ \end{align*}