Internal problem ID [1864]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of
parameters. Page 366
Problem number: 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )+3 x_{2} \relax (t )+2 x_{3} \relax (t )+\sin \relax (t )\\ x_{2}^{\prime }\relax (t )&=-x_{1} \relax (t )+2 x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=4 x_{1} \relax (t )-x_{2} \relax (t )-x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 84
dsolve([diff(x__1(t),t)=1*x__1(t)+3*x__2(t)+2*x__3(t)+sin(t),diff(x__2(t),t)=-1*x__1(t)+2*x__2(t)+1*x__3(t),diff(x__3(t),t)=4*x__1(t)-1*x__2(t)-1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = -\frac {\sin \relax (t )}{10}-\frac {\cos \relax (t )}{5}+\frac {c_{1} {\mathrm e}^{t}}{3}-\frac {{\mathrm e}^{-2 t} c_{2}}{3}+c_{3} {\mathrm e}^{3 t} \] \[ x_{2} \relax (t ) = \frac {\cos \relax (t )}{10}+\frac {3 \sin \relax (t )}{10}-\frac {2 c_{1} {\mathrm e}^{t}}{3}-\frac {{\mathrm e}^{-2 t} c_{2}}{3} \] \[ x_{3} \relax (t ) = -\frac {\cos \relax (t )}{10}-\frac {4 \sin \relax (t )}{5}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-2 t} c_{2}+c_{3} {\mathrm e}^{3 t} \]
✓ Solution by Mathematica
Time used: 0.207 (sec). Leaf size: 187
DSolve[{x1'[t]==1*x1[t]+3*x2[t]+2*x3[t]+Sin[t],x2'[t]==-1*x1[t]+2*x2[t]+1*x3[t],x3'[t]==4*x1[t]-1*x2[t]-1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{30} \left (-3 \sin (t)-6 \cos (t)+5 e^{-2 t} \left ((c_1-4 c_2-c_3) e^{3 t}+3 (c_1+2 c_2+c_3) e^{5 t}+2 c_1-2 (c_2+c_3)\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} (c_1-c_2-c_3) e^{-2 t}+\frac {1}{3} (-c_1+4 c_2+c_3) e^t+\frac {1}{10} (3 \sin (t)+\cos (t)) \\ \text {x3}(t)\to \frac {1}{10} \left (-8 \sin (t)-\cos (t)+5 (c_1-4 c_2-c_3) e^t+10 (-c_1+c_2+c_3) e^{-2 t}+5 (c_1+2 c_2+c_3) e^{3 t}\right ) \\ \end{align*}