Internal problem ID [1865]
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: 13.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )+2 x_{2} \relax (t )-3 x_{3} \relax (t )+{\mathrm e}^{t}\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t )+2 x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )-x_{2} \relax (t )+4 x_{3} \relax (t )-{\mathrm e}^{t} \end {align*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 123
dsolve([diff(x__1(t),t)=1*x__1(t)+2*x__2(t)-3*x__3(t)+exp(t),diff(x__2(t),t)=1*x__1(t)+1*x__2(t)+2*x__3(t),diff(x__3(t),t)=1*x__1(t)-1*x__2(t)+4*x__3(t)-exp(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = 2 \,{\mathrm e}^{t}-c_{1} {\mathrm e}^{2 t}-c_{2} {\mathrm e}^{2 t} t +c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{2 t} c_{3} t^{2}+2 \,{\mathrm e}^{2 t} c_{3} t +2 c_{3} {\mathrm e}^{2 t} \] \[ x_{2} \relax (t ) = -2 \,{\mathrm e}^{t}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{2 t} t +{\mathrm e}^{2 t} c_{3} t^{2}+2 c_{3} {\mathrm e}^{2 t} \] \[ x_{3} \relax (t ) = -{\mathrm e}^{t}+c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{2 t} t +{\mathrm e}^{2 t} c_{3} t^{2} \]
✓ Solution by Mathematica
Time used: 0.084 (sec). Leaf size: 122
DSolve[{x1'[t]==1*x1[t]+2*x2[t]-3*x3[t]+Exp[t],x2'[t]==1*x1[t]+1*x2[t]+2*x3[t],x3'[t]==1*x1[t]-1*x2[t]+4*x3[t]-Exp[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{2} e^t \left (4+e^t (-2 c_1 (t-1)-c_2 (t-4) t+c_3 (t-6) t)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^t \left (-4+e^t (t (c_2 (t-2)-c_3 (t-4)+2 c_1)+2 c_2)\right ) \\ \text {x3}(t)\to \frac {1}{2} e^t \left (-2+e^t (2 c_1 t+c_2 (t-2) t+c_3 (2-(t-4) t))\right ) \\ \end{align*}