Internal problem ID [12382]
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(c).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )-2 y+4 z \left (t \right )\\ y^{\prime }&=-2 x \left (t \right )+y+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x \left (t \right )-2 y+6 z \left (t \right )+{\mathrm e}^{2 t} \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 77
dsolve([diff(x(t),t)=-2*x(t)-2*y(t)+4*z(t),diff(y(t),t)=-2*x(t)+1*y(t)+2*z(t),diff(z(t),t)=-4*x(t)-2*y(t)+6*z(t)+exp(2*t)],singsol=all)
\begin{align*} x \left (t \right ) &= \left ({\mathrm e}^{t} \left (4 t +c_{2} -4\right )+c_{3} \right ) {\mathrm e}^{t} \\ y \left (t \right ) &= \left (\frac {c_{2}}{2}+2 t -2+c_{1} \right ) {\mathrm e}^{2 t}+\frac {c_{3} {\mathrm e}^{t}}{2} \\ z \left (t \right ) &= \frac {5 c_{2} {\mathrm e}^{2 t}}{4}+5 \,{\mathrm e}^{2 t} t -4 \,{\mathrm e}^{2 t}+c_{3} {\mathrm e}^{t}+\frac {c_{1} {\mathrm e}^{2 t}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 118
DSolve[{x'[t]==-2*x[t]-2*y[t]+4*z[t],y'[t]==-2*x[t]+y[t]+2*z[t],z'[t]==-4*x[t]-2*y[t]+6*z[t]+Exp[2*t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^t \left (e^t (4 t-4-3 c_1-2 c_2+4 c_3)+2 (2 c_1+c_2-2 c_3)\right ) \\ y(t)\to e^t \left (2 e^t (t-1-c_1+c_3)+2 c_1+c_2-2 c_3\right ) \\ z(t)\to e^t \left (e^t (5 t-4-4 c_1-2 c_2+5 c_3)+2 (2 c_1+c_2-2 c_3)\right ) \\ \end{align*}