Internal problem ID [12389]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters).
Problems page 514
Problem number: Problem 6(b).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y-z \left (t \right )+5 \sin \left (t \right )\\ y^{\prime }&=y+z \left (t \right )-10 \cos \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )+2 \end {align*}
With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2, z \left (0\right ) = 3] \]
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 71
dsolve([diff(x(t),t) = 2*x(t)+y(t)-z(t)+5*sin(t), diff(y(t),t) = y(t)+z(t)-10*cos(t), diff(z(t),t) = x(t)+z(t)+2, x(0) = 1, y(0) = 2, z(0) = 3], singsol=all)
\begin{align*} x \left (t \right ) &= -2 \cos \left (t \right )-1-3 \,{\mathrm e}^{t} \sin \left (t \right )+4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ y \left (t \right ) &= -4 \sin \left (t \right )+5 \cos \left (t \right )+1+3 \,{\mathrm e}^{t} \sin \left (t \right )-4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ z \left (t \right ) &= -1-\sin \left (t \right )+\cos \left (t \right )+3 \,{\mathrm e}^{t} \cos \left (t \right )+4 \,{\mathrm e}^{t} \sin \left (t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.398 (sec). Leaf size: 74
DSolve[{x'[t]==2*x[t]+y[t]-z[t]+5*Sin[t],y'[t]==y[t]+z[t]-10*Cos[t],z'[t]==x[t]+z[t]+2},{x[0]==1,y[0]==2,z[0]==3},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -3 e^t \sin (t)+\left (4 e^t-2\right ) \cos (t)-1 \\ y(t)\to \left (3 e^t-4\right ) \sin (t)+\left (5-4 e^t\right ) \cos (t)+1 \\ z(t)\to \left (4 e^t-1\right ) \sin (t)+\left (3 e^t+1\right ) \cos (t)-1 \\ \end{align*}