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 }\left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+2 x_{3} \left (t \right )+\sin \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )-x_{3} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.25 (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)
\[ x_{1} \left (t \right ) = -\frac {\sin \left (t \right )}{10}-\frac {\cos \left (t \right )}{5}+\frac {c_{1} {\mathrm e}^{t}}{3}-\frac {{\mathrm e}^{-2 t} c_{2}}{3}+c_{3} {\mathrm e}^{3 t} \] \[ x_{2} \left (t \right ) = \frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}-\frac {2 c_{1} {\mathrm e}^{t}}{3}-\frac {{\mathrm e}^{-2 t} c_{2}}{3} \] \[ x_{3} \left (t \right ) = -\frac {\cos \left (t \right )}{10}-\frac {4 \sin \left (t \right )}{5}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-2 t} c_{2} +c_{3} {\mathrm e}^{3 t} \]
✓ Solution by Mathematica
Time used: 0.318 (sec). Leaf size: 211
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 \left (e^{3 t}+3 e^{5 t}+2\right )+c_2 \left (-4 e^{3 t}+6 e^{5 t}-2\right )+c_3 \left (-e^{3 t}+3 e^{5 t}-2\right )\right )\right ) \text {x2}(t)\to \frac {1}{30} \left (9 \sin (t)+3 \cos (t)-10 e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )-4 c_2 e^{3 t}-c_3 e^{3 t}+c_2+c_3\right )\right ) \text {x3}(t)\to \frac {1}{10} \left (-8 \sin (t)-\cos (t)+5 e^{-2 t} \left (c_1 \left (e^{3 t}+e^{5 t}-2\right )+2 c_2 \left (-2 e^{3 t}+e^{5 t}+1\right )+c_3 \left (-e^{3 t}+e^{5 t}+2\right )\right )\right ) \end{align*}