Internal problem ID [1861]
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: 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-x_{1} \relax (t )-x_{2} \relax (t )-2 x_{3} \relax (t )+{\mathrm e}^{t}\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )+3 x_{3} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 0, x_{2} \relax (0) = 0, x_{3} \relax (0) = 0] \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 47
dsolve([diff(x__1(t),t) = -x__1(t)-x__2(t)-2*x__3(t)+exp(t), diff(x__2(t),t) = x__1(t)+x__2(t)+x__3(t), diff(x__3(t),t) = 2*x__1(t)+x__2(t)+3*x__3(t), x__1(0) = 0, x__2(0) = 0, x__3(0) = 0],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = -\frac {{\mathrm e}^{t} \left (t^{3}+6 t^{2}-6 t \right )}{6} \] \[ x_{2} \relax (t ) = \frac {{\mathrm e}^{t} t^{2}}{2} \] \[ x_{3} \relax (t ) = \frac {{\mathrm e}^{t} \left (t^{3}+6 t^{2}\right )}{6} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 49
DSolve[{x1'[t]==-1*x1[t]-1*x2[t]-2*x3[t]+Exp[t],x2'[t]==1*x1[t]+1*x2[t]+1*x3[t],x3'[t]==2*x1[t]+1*x2[t]+3*x3[t]},{x1[0]==0,x2[0]==0,x3[0]==0},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -\frac {1}{6} e^t t (t (t+6)-6) \\ \text {x2}(t)\to \frac {e^t t^2}{2} \\ \text {x3}(t)\to \frac {1}{6} e^t t^2 (t+6) \\ \end{align*}