Internal problem ID [11043]
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(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y \left (t \right )-3 z \left (t \right )+2 \,{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-y \left (t \right )+2 z \left (t \right )+4 \,{\mathrm e}^{t}\\ z^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )+3 z \left (t \right )+4 \,{\mathrm e}^{t} \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: 0.047 (sec). Leaf size: 86
dsolve([diff(x(t),t) = -3*x(t)+y(t)-3*z(t)+2*exp(t), diff(y(t),t) = 4*x(t)-y(t)+2*z(t)+4*exp(t), diff(z(t),t) = 4*x(t)-2*y(t)+3*z(t)+4*exp(t), x(0) = 1, y(0) = 2, z(0) = 3],[x(t), y(t), z(t)], singsol=all)
\[ x \left (t \right ) = -\frac {3 \,{\mathrm e}^{t}}{2}-2 \,{\mathrm e}^{-t} \sin \left (2 t \right )+\frac {5 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \] \[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \] \[ z \left (t \right ) = \frac {7 \,{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 98
DSolve[{x'[t]==-3*x[t]+y[t]-3*z[t]+2*Exp[t],y'[t]==4*x[t]-y[t]+2*z[t]+4*Exp[t],z'[t]==4*x[t]-2*y[t]+3*z[t]+4*Exp[t]},{x[0]==1,y[0]==2,z[0]==3},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {1}{2} e^{-t} \left (3 e^{2 t}+4 \sin (2 t)-5 \cos (2 t)\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (5 e^{2 t}+9 \sin (2 t)-\cos (2 t)\right ) \\ z(t)\to \frac {1}{2} e^{-t} \left (7 e^{2 t}+9 \sin (2 t)-\cos (2 t)\right ) \\ \end{align*}