Internal problem ID [12380]
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 4(a).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y-2 z \left (t \right )-2 \sinh \left (t \right )\\ y^{\prime }&=4 x \left (t \right )+2 y-2 z \left (t \right )+10 \cosh \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y+z \left (t \right )+5 \end {align*}
✓ Solution by Maple
Time used: 0.282 (sec). Leaf size: 429
dsolve([diff(x(t),t)=2*x(t)+4*y(t)-2*z(t)-2*sinh(t),diff(y(t),t)=4*x(t)+2*y(t)-2*z(t)+10*cosh(t),diff(z(t),t)=-x(t)+3*y(t)+z(t)+5],singsol=all)
\begin{align*} x \left (t \right ) &= -1+2 c_{3} {\mathrm e}^{5 t}+\frac {9 c_{1} {\mathrm e}^{-2 t}}{8}-\frac {45 \cosh \left (t \right )}{16}-\frac {3 \sinh \left (t \right )}{16}+\frac {c_{2} {\mathrm e}^{2 t}}{2}-\frac {275 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{224}-\frac {3 \,{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{14}-\frac {275 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{224}-\frac {3 \,{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{14}+\frac {3 \,{\mathrm e}^{2 t} \sinh \left (t \right )}{2}+\frac {275 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{288}-\frac {3 \,{\mathrm e}^{2 t} \cosh \left (t \right )}{2}-\frac {275 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{288}-\frac {3 \,{\mathrm e}^{5 t} \sinh \left (4 t \right )}{14}-\frac {275 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{1008}+\frac {3 \,{\mathrm e}^{5 t} \cosh \left (4 t \right )}{14}+\frac {275 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{1008} \\ y \left (t \right ) &= -1+2 c_{3} {\mathrm e}^{5 t}-\frac {5 c_{1} {\mathrm e}^{-2 t}}{8}-\frac {15 \cosh \left (t \right )}{16}-\frac {\sinh \left (t \right )}{16}+\frac {c_{2} {\mathrm e}^{2 t}}{2}+\frac {25 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{32}-\frac {{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{14}+\frac {25 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{32}-\frac {{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{14}+\frac {{\mathrm e}^{2 t} \sinh \left (t \right )}{2}-\frac {175 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{288}-\frac {{\mathrm e}^{2 t} \cosh \left (t \right )}{2}+\frac {175 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{288}-\frac {{\mathrm e}^{5 t} \sinh \left (4 t \right )}{14}+\frac {25 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{144}+\frac {{\mathrm e}^{5 t} \cosh \left (4 t \right )}{14}-\frac {25 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{144} \\ z \left (t \right ) &= -\frac {25 \,{\mathrm e}^{-2 t} \sinh \left (t \right )}{14}-3-\frac {4 \,{\mathrm e}^{-2 t} \sinh \left (3 t \right )}{7}-\frac {25 \,{\mathrm e}^{-2 t} \cosh \left (t \right )}{14}-\frac {4 \,{\mathrm e}^{-2 t} \cosh \left (3 t \right )}{7}+4 \,{\mathrm e}^{2 t} \sinh \left (t \right )+\frac {25 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{18}-4 \,{\mathrm e}^{2 t} \cosh \left (t \right )-\frac {25 \,{\mathrm e}^{2 t} \cosh \left (3 t \right )}{18}-\frac {4 \,{\mathrm e}^{5 t} \sinh \left (4 t \right )}{7}-\frac {25 \,{\mathrm e}^{5 t} \sinh \left (6 t \right )}{63}+\frac {4 \,{\mathrm e}^{5 t} \cosh \left (4 t \right )}{7}+\frac {25 \,{\mathrm e}^{5 t} \cosh \left (6 t \right )}{63}+c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{5 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.292 (sec). Leaf size: 233
DSolve[{x'[t]==2*x[t]+4*y[t]-2*z[t]-2*Sinh[t],y'[t]==4*x[t]+2*y[t]-2*z[t]+10*Cosh[t],z'[t]==-x[t]+3*y[t]+z[t]+5},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {29 e^{-t}}{9}-3 e^t+\frac {9}{14} (c_1-c_2) e^{-2 t}+\frac {2}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{6} (-3 c_1+c_2+4 c_3) e^{2 t}-1 \\ y(t)\to \frac {7 e^{-t}}{9}-e^t+\frac {5}{14} (c_2-c_1) e^{-2 t}+\frac {2}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{6} (-3 c_1+c_2+4 c_3) e^{2 t}-1 \\ z(t)\to -\frac {25 e^{-t}}{9}-4 e^t+\frac {4}{7} (c_1-c_2) e^{-2 t}+\frac {1}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{3} (-3 c_1+c_2+4 c_3) e^{2 t}-3 \\ \end{align*}