76.25.17 problem 21

Internal problem ID [17830]
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 : 21
Date solved : Tuesday, January 28, 2025 at 11:03:33 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.205 (sec). Leaf size: 80

dsolve([diff(x__1(t),t)=-2*x__1(t)+2*x__2(t)+0*x__3(t)-2*x__4(t),diff(x__2(t),t)=-1*x__1(t)+3*x__2(t)-1*x__3(t)+1*x__4(t),diff(x__3(t),t)=-2*x__1(t)-2*x__2(t)-4*x__3(t)+2*x__4(t),diff(x__4(t),t)=-7*x__1(t)+1*x__2(t)-7*x__3(t)+3*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 241

DSolve[{D[x1[t],t]==-2*x1[t]+2*x2[t]+0*x3[t]-2*x4[t],D[x2[t],t]==-1*x1[t]+3*x2[t]-1*x3[t]+1*x4[t],D[x3[t],t]==-2*x1[t]-2*x2[t]-4*x3[t]+2*x4[t],D[x4[t],t]==-7*x1[t]+1*x2[t]-7*x3[t]+3*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-4 t} \left (c_1 \left (-e^{2 t}+e^{6 t}+2\right )+(c_2+c_3-c_4) e^{6 t}+(-c_2-3 c_3+c_4) e^{2 t}+2 c_3\right ) \\ \text {x2}(t)\to -\frac {1}{2} e^{2 t} \left (c_1 \left (e^{2 t}-1\right )-c_2 \left (e^{2 t}+1\right )+(c_3-c_4) \left (e^{2 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-2 t} \left (c_1 \left (-e^{4 t}\right )-c_2 e^{4 t}-c_3 e^{4 t}+c_4 e^{4 t}+c_1+c_2+3 c_3-c_4\right ) \\ \text {x4}(t)\to \frac {1}{2} e^{-4 t} \left (-\left (c_1 \left (e^{6 t}+e^{8 t}-2\right )\right )-(c_2+c_3-c_4) e^{6 t}+(c_2-c_3+c_4) e^{8 t}+2 c_3\right ) \\ \end{align*}