76.28.8 problem 9

Internal problem ID [17875]
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 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 11:04:23 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.592 (sec). Leaf size: 81

dsolve([diff(x__1(t),t)=-1/2*x__1(t)+1*x__2(t)+1/2*x__3(t),diff(x__2(t),t)=1*x__1(t)-1*x__2(t)+1*x__3(t)-sin(t),diff(x__3(t),t)=1/2*x__1(t)+1*x__2(t)-1/2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -\frac {\sin \left (t \right )}{10}+\frac {3 \cos \left (t \right )}{10}+c_{1} {\mathrm e}^{t}-2 \,{\mathrm e}^{-2 t} c_{2} \\ x_{3} \left (t \right ) &= \frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-2 t} c_{2} -c_{3} {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 194

DSolve[{D[x1[t],t]==-1/2*x1[t]+1*x2[t]+1/2*x3[t]+0,D[x2[t],t]==1*x1[t]-1*x2[t]+1*x3[t]-Sin[t],D[x3[t],t]==1/2*x1[t]+1*x2[t]-1/2*x3[t]+0},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{30} \left (9 \sin (t)+3 \cos (t)+5 e^{-2 t} \left (c_1 \left (3 e^t+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (-3 e^t+2 e^{3 t}+1\right )\right )\right ) \\ \text {x2}(t)\to \frac {1}{30} \left (-3 \sin (t)+9 \cos (t)+10 e^{-2 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 )\right ) \\ \text {x3}(t)\to \frac {1}{30} \left (9 \sin (t)+3 \cos (t)+5 e^{-2 t} \left (c_1 \left (-3 e^t+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (3 e^t+2 e^{3 t}+1\right )\right )\right ) \\ \end{align*}