14.26.14 problem 14

Internal problem ID [2787]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 14
Date solved : Monday, January 27, 2025 at 06:13:51 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.225 (sec). Leaf size: 59

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

\begin{align*} x_{1} \left (t \right ) &= 0 \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-2 \sin \left (2 t \right )-t \sin \left (2 t \right )+2 \cos \left (2 t \right )\right )}{2} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (5 \sin \left (2 t \right )+4 \cos \left (2 t \right )+2 t \cos \left (2 t \right )\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 58

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

\begin{align*} \text {x1}(t)\to 0 \\ \text {x2}(t)\to \frac {1}{2} e^t (2 \cos (2 t)-(t+2) \sin (2 t)) \\ \text {x3}(t)\to \frac {1}{4} e^t (5 \sin (2 t)+2 (t+2) \cos (2 t)) \\ \end{align*}