Internal problem ID [12390]
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(c).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+3 y+z \left (t \right )+10 \cos \left (t \right ) \sin \left (t \right )\\ y^{\prime }&=x \left (t \right )-5 y-3 z \left (t \right )+10 \cos \left (t \right )^{2}-5\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+7 y+3 z \left (t \right )+23 \,{\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: 0.829 (sec). Leaf size: 132
dsolve([diff(x(t),t) = -3*x(t)+3*y(t)+z(t)+5*sin(2*t), diff(y(t),t) = x(t)-5*y(t)-3*z(t)+5*cos(2*t), diff(z(t),t) = -3*x(t)+7*y(t)+3*z(t)+23*exp(t), x(0) = 1, y(0) = 2, z(0) = 3], singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {69 \,{\mathrm e}^{t}}{26}+\sin \left (2 t \right )+\frac {\cos \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {191 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{13} \\ y \left (t \right ) &= -\frac {253 \,{\mathrm e}^{t}}{26}-\frac {5 \sin \left (2 t \right )}{2}+\frac {21 \,{\mathrm e}^{-t}}{2}+\frac {191 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26}+\frac {16 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{13} \\ z \left (t \right ) &= \frac {483 \,{\mathrm e}^{t}}{26}+\frac {7 \cos \left (2 t \right )}{2}+\frac {9 \sin \left (2 t \right )}{2}-\frac {21 \,{\mathrm e}^{-t}}{2}-\frac {223 \,{\mathrm e}^{-2 t} \cos \left (2 t \right )}{26}-\frac {159 \,{\mathrm e}^{-2 t} \sin \left (2 t \right )}{26} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.393 (sec). Leaf size: 197
DSolve[{x'[t]==-3*x[t]+3*y[t]+z[t]+5*Sin[3*t],y'[t]==x[t]-5*y[t]-3*z[t]+5*Cos[2*t],z'[t]==-3*x[t]+7*y[t]+3*z[t]+23*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}{754} \left (7540 e^{-t}-2001 e^t+603 e^{-2 t} \sin (2 t)+377 \sin (2 t)+429 \sin (3 t)+\left (1131-5409 e^{-2 t}\right ) \cos (2 t)-507 \cos (3 t)\right ) \\ y(t)\to \frac {1}{754} \left (7540 e^{-t}-7337 e^t+5409 e^{-2 t} \sin (2 t)-1508 \sin (2 t)-507 \sin (3 t)+\left (603 e^{-2 t}+1131\right ) \cos (2 t)-429 \cos (3 t)\right ) \\ z(t)\to -10 e^{-t}+\frac {483 e^t}{26}-\frac {2403}{377} e^{-2 t} \sin (2 t)+\frac {43}{58} \sin (3 t)+\left (1-\frac {3006 e^{-2 t}}{377}\right ) \cos (2 t)+\frac {81}{58} \cos (3 t)+9 \sin (t) \cos (t) \\ \end{align*}