Internal problem ID [1856]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=4 x_{1} \relax (t )+5 x_{2} \relax (t )+4 \,{\mathrm e}^{t} \cos \relax (t )\\ x_{2}^{\prime }\relax (t )&=-2 x_{1} \relax (t )-2 x_{2} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 0, x_{2} \relax (0) = 0] \]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 34
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],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {{\mathrm e}^{t} \left (12 \sin \relax (t ) t +4 \cos \relax (t ) t +4 \sin \relax (t )\right )}{2} \] \[ x_{2} \relax (t ) = -4 \,{\mathrm e}^{t} \sin \relax (t ) t \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 33
DSolve[{x1'[t]==4*x1[t]+5*x2[t]+4*Exp[t]*Cos[t],x2'[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*}