Internal problem ID [1866]
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: 14.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-x_{1} \relax (t )-x_{2} \relax (t )+1\\ x_{2}^{\prime }\relax (t )&=-4 x_{2} \relax (t )-x_{3} \relax (t )+t\\ x_{3}^{\prime }\relax (t )&=5 x_{2} \relax (t )+{\mathrm e}^{t} \end {align*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 124
dsolve([diff(x__1(t),t)=-1*x__1(t)-1*x__2(t)+0*x__3(t)+1,diff(x__2(t),t)=0*x__1(t)-4*x__2(t)-1*x__3(t)+t,diff(x__3(t),t)=0*x__1(t)+5*x__2(t)-0*x__3(t)+exp(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {c_{2} {\mathrm e}^{-2 t} \sin \relax (t )}{10}-\frac {3 \,{\mathrm e}^{-2 t} \sin \relax (t ) c_{3}}{10}-\frac {3 c_{2} {\mathrm e}^{-2 t} \cos \relax (t )}{10}-\frac {{\mathrm e}^{-2 t} \cos \relax (t ) c_{3}}{10}+\frac {{\mathrm e}^{t}}{20}+\frac {4}{5}+{\mathrm e}^{-t} c_{1} \] \[ x_{2} \relax (t ) = -\frac {2 \,{\mathrm e}^{-2 t} \sin \relax (t ) c_{3}}{5}+\frac {{\mathrm e}^{-2 t} \cos \relax (t ) c_{3}}{5}-\frac {2 c_{2} {\mathrm e}^{-2 t} \cos \relax (t )}{5}-\frac {c_{2} {\mathrm e}^{-2 t} \sin \relax (t )}{5}-\frac {{\mathrm e}^{t}}{10}+\frac {1}{5} \] \[ x_{3} \relax (t ) = {\mathrm e}^{-2 t} \sin \relax (t ) c_{3}+c_{2} {\mathrm e}^{-2 t} \cos \relax (t )+\frac {{\mathrm e}^{t}}{2}+t -\frac {4}{5} \]
✓ Solution by Mathematica
Time used: 0.528 (sec). Leaf size: 135
DSolve[{x1'[t]==-1*x1[t]-1*x2[t]+0*x3[t]+1,x2'[t]==0*x1[t]-4*x2[t]-1*x3[t]+t,x3'[t]==0*x1[t]+5*x2[t]-0*x3[t]+Exp[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{20} e^{-2 t} \left (e^t \left (e^t \left (e^t+16\right )+10 (2 c_1+c_2+c_3)\right )-10 (c_2+c_3) \cos (t)-10 (3 c_2+c_3) \sin (t)\right ) \\ \text {x2}(t)\to -\frac {e^t}{10}+e^{-2 t} (c_2 \cos (t)-(2 c_2+c_3) \sin (t))+\frac {1}{5} \\ \text {x3}(t)\to t+\frac {e^t}{2}+e^{-2 t} (5 c_2 \sin (t)+c_3 (2 \sin (t)+\cos (t)))-\frac {4}{5} \\ \end{align*}