76.25.18 problem 22

Internal problem ID [17831]
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 : 22
Date solved : Tuesday, January 28, 2025 at 11:03:34 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.168 (sec). Leaf size: 101

dsolve([diff(x__1(t),t)=-5*x__1(t)-2*x__2(t)-1*x__3(t)+2*x__4(t)+3*x__5(t),diff(x__2(t),t)=0*x__1(t)-3*x__2(t)-0*x__3(t)+0*x__4(t)+0*x__5(t),diff(x__3(t),t)=1*x__1(t)-0*x__2(t)-1*x__3(t)+0*x__4(t)-1*x__5(t),diff(x__4(t),t)=2*x__1(t)+1*x__2(t)-0*x__3(t)-4*x__4(t)-2*x__5(t),diff(x__5(t),t)=-3*x__1(t)-2*x__2(t)-1*x__3(t)+2*x__4(t)+1*x__5(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= c_5 \,{\mathrm e}^{-3 t} \\ x_{3} \left (t \right ) &= -c_{3} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{-2 t} \\ x_{4} \left (t \right ) &= c_5 \,{\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{-4 t}-c_{1} {\mathrm e}^{-2 t} \\ x_{5} \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 245

DSolve[{D[x1[t],t]==-5*x1[t]-2*x2[t]-1*x3[t]+2*x4[t]+3*x5[t],D[x2[t],t]==0*x1[t]-3*x2[t]-0*x3[t]+0*x4[t]+0*x5[t],D[x3[t],t]==1*x1[t]-0*x2[t]-1*x3[t]+0*x4[t]-x5[t],D[x4[t],t]==2*x1[t]+1*x2[t]-0*x3[t]-4*x4[t]-2*x5[t],D[x5[t],t]==-3*x1[t]-2*x2[t]-1*x3[t]+2*x4[t]+1*x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-4 t} \left (c_1 \left (e^{2 t}-e^{3 t}+1\right )-c_2 e^{2 t}+c_3 e^{2 t}-c_3 e^{3 t}+c_4 e^{2 t}+c_5 e^{3 t}+c_2-c_4-c_5\right ) \\ \text {x2}(t)\to c_2 e^{-3 t} \\ \text {x3}(t)\to e^{-2 t} \left (c_1 \left (e^t-1\right )+(c_3-c_5) e^t+c_5\right ) \\ \text {x4}(t)\to e^{-4 t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^t-1\right )-c_5 e^{2 t}+c_4+c_5\right ) \\ \text {x5}(t)\to e^{-4 t} \left (c_1 \left (-e^{3 t}\right )-c_2 e^{2 t}+c_3 e^{2 t}-c_3 e^{3 t}+c_4 e^{2 t}+c_5 e^{2 t}+c_5 e^{3 t}+c_1+c_2-c_4-c_5\right ) \\ \end{align*}