Internal problem ID [1868]
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: 17.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1}\relax (t )-x_{2}\relax (t )-x_{3}\relax (t )+{\mathrm e}^{3 t}\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )+3 x_{2}\relax (t )+x_{3}\relax (t )-{\mathrm e}^{3 t}\\ x_{3}^{\prime }\relax (t )&=-3 x_{1}\relax (t )+x_{2}\relax (t )-x_{3}\relax (t )-{\mathrm e}^{3 t} \end {align*}
✓ Solution by Maple
Time used: 0.404 (sec). Leaf size: 86
dsolve([diff(x__1(t),t)=1*x__1(t)-1*x__2(t)-1*x__3(t)+exp(3*t),diff(x__2(t),t)=1*x__1(t)+3*x__2(t)+1*x__3(t)-1*exp(3*t),diff(x__3(t),t)=-3*x__1(t)+1*x__2(t)-1*x__3(t)-exp(3*t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = {\mathrm e}^{3 t} t +\frac {c_{1} {\mathrm e}^{-2 t}}{4}-c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{3 t} c_{3} \] \[ x_{2}\relax (t ) = -{\mathrm e}^{3 t} t -\frac {c_{1} {\mathrm e}^{-2 t}}{4}+{\mathrm e}^{3 t} c_{3} \] \[ x_{3}\relax (t ) = -{\mathrm e}^{3 t} t +c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{3 t} c_{3} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 139
DSolve[{x1'[t]==1*x1[t]-1*x2[t]-1*x3[t]+Exp[3*t],x2'[t]==1*x1[t]+3*x2[t]+1*x3[t]-Exp[3*t],x3'[t]==-3*x1[t]+1*x2[t]-1*x3[t]-Exp[3*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-2 t} \left (5 (c_1+c_2) e^{4 t}-e^{5 t} (-5 t+c_1+5 c_2+c_3)+c_1+c_3\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-2 t} \left (e^{5 t} (-5 t+c_1+5 c_2+c_3)-c_1-c_3\right ) \\ \text {x3}(t)\to \frac {1}{5} e^{-2 t} \left (-5 (c_1+c_2) e^{4 t}+e^{5 t} (-5 t+c_1+5 c_2+c_3)+4 (c_1+c_3)\right ) \\ \end{align*}