14.26.7 problem 7

Internal problem ID [2780]
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 : 7
Date solved : Monday, January 27, 2025 at 06:13:46 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 45

dsolve([diff(x__1(t),t) = 4*x__1(t)+5*x__2(t)+4*exp(t)*cos(t), diff(x__2(t),t) = -2*x__1(t)-2*x__2(t), x__1(0) = 1, x__2(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (20 \sin \left (t \right )+12 \sin \left (t \right ) t +2 \cos \left (t \right )+4 \cos \left (t \right ) t \right )}{2} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (\cos \left (t \right )-5 \sin \left (t \right )-4 \sin \left (t \right ) t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 46

DSolve[{D[x1[t],t]==4*x1[t]+5*x2[t]+4*Exp[t]*Cos[t],D[x2[t],t]==-2*x1[t]-2*x2[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (2 (3 t+5) \sin (t)+(2 t+1) \cos (t)) \\ \text {x2}(t)\to e^t (\cos (t)-(4 t+5) \sin (t)) \\ \end{align*}