Internal problem ID [791]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number: 18.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-2 x_{1}\relax (t )+x_{2}\relax (t )+2 \,{\mathrm e}^{-t}\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )-2 x_{2}\relax (t )+3 t \end {align*}
With initial conditions \[ [x_{1}\relax (0) = \alpha _{1}, x_{2}\relax (0) = \alpha _{2}] \]
✓ Solution by Maple
Time used: 0.048 (sec). Leaf size: 93
dsolve([diff(x__1(t),t) = -2*x__1(t)+x__2(t)+2*exp(-t), diff(x__2(t),t) = x__1(t)-2*x__2(t)+3*t, x__1(0) = alpha__1, x__2(0) = alpha__2],[x__1(t), x__2(t)], singsol=all)
\[ x_{1}\relax (t ) = \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-t}-\left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-3 t}+\frac {{\mathrm e}^{-t}}{2}+t \,{\mathrm e}^{-t}-\frac {4}{3}+t \] \[ x_{2}\relax (t ) = \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-t}+\left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right ) {\mathrm e}^{-3 t}+t \,{\mathrm e}^{-t}+2 t -\frac {5}{3}-\frac {{\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 94
DSolve[{x1'[t]==-2*x1[t]+1*x2[t]+2*Exp[-t],x2'[t]==1*x1[t]-2*x2[t]+3*t},{x1[0]==a1,x2[0]==a2},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{-3 t} \left (3 e^{2 t} (\text {a1}+\text {a2}+2 t+4)+3 \text {a1}-3 \text {a2}+2 e^{3 t} (3 t-4)-4\right ) \\ \text {x2}(t)\to \frac {1}{6} e^{-3 t} \left (3 e^{2 t} (\text {a1}+\text {a2}+2 t+2)-3 \text {a1}+3 \text {a2}+2 e^{3 t} (6 t-5)+4\right ) \\ \end{align*}