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 }\left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )+1\\ x_{2}^{\prime }\left (t \right )&=-4 x_{2} \left (t \right )-x_{3} \left (t \right )+t\\ x_{3}^{\prime }\left (t \right )&=5 x_{2} \left (t \right )+{\mathrm e}^{t} \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 123
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)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )}{2}+\frac {{\mathrm e}^{-2 t} \sin \left (t \right ) c_{3}}{2}+\frac {c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )}{2}+\frac {{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3}}{2}+\frac {{\mathrm e}^{t}}{20}+\frac {4}{5}+{\mathrm e}^{-t} c_{1} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} \sin \left (t \right ) c_{3} +c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )+\frac {1}{5}-\frac {{\mathrm e}^{t}}{10} \\ x_{3} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_{3} -{\mathrm e}^{-2 t} \cos \left (t \right ) c_{3} -2 c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )+c_{2} {\mathrm e}^{-2 t} \sin \left (t \right )+\frac {{\mathrm e}^{t}}{2}-\frac {4}{5}+t \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.816 (sec). Leaf size: 144
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 (16 e^t+e^{2 t}+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 {1}{10} \left (2-e^t\right )+c_2 e^{-2 t} \cos (t)-(2 c_2+c_3) e^{-2 t} \sin (t) \\ \text {x3}(t)\to t+\frac {e^t}{2}+c_3 e^{-2 t} \cos (t)+(5 c_2+2 c_3) e^{-2 t} \sin (t)-\frac {4}{5} \\ \end{align*}