14.25.3 problem 1

Internal problem ID [2760]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 1
Date solved : Monday, January 27, 2025 at 06:12:53 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 ) = 0\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 33

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) = 0, x__2(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 33

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]==0,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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