76.28.7 problem 8

Internal problem ID [17874]
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 : 8
Date solved : Tuesday, January 28, 2025 at 11:04:22 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 )+3 x_{3} \left (t \right )+3 t\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )+3 \cos \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.641 (sec). Leaf size: 75

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

\begin{align*} x_{1} \left (t \right ) &= -c_{1} {\mathrm e}^{-2 t}+\frac {15}{4}-\frac {3 t}{2}+\frac {9 \cos \left (t \right )}{10}+\frac {27 \sin \left (t \right )}{10}+c_{2} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_{3} {\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= -\frac {2 c_{1} {\mathrm e}^{-2 t}}{3}+\frac {9}{2}+\frac {33 \sin \left (t \right )}{10}+\frac {21 \cos \left (t \right )}{10}+c_{2} {\mathrm e}^{-t}-3 t -\frac {c_{3} {\mathrm e}^{-2 t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.229 (sec). Leaf size: 138

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

\begin{align*} \text {x1}(t)\to -\frac {3}{4} (2 t-5)+\frac {27 \sin (t)}{10}+\frac {9 \cos (t)}{10}+(3 c_1-c_2-3 c_3) e^{-2 t}+(-2 c_1+c_2+3 c_3) e^{-t} \\ \text {x2}(t)\to c_2 e^{-2 t} \\ \text {x3}(t)\to \frac {1}{10} \left (33 \sin (t)+21 \cos (t)-5 e^{-2 t} \left (e^{2 t} (6 t-9)+(4 c_1-2 c_2-6 c_3) e^t-4 c_1+2 c_2+4 c_3\right )\right ) \\ \end{align*}