Internal problem ID [1857]
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: 2.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )+{\mathrm e}^{t} \end {align*}
With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 1] \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 32
dsolve([diff(x__1(t),t) = 3*x__1(t)-4*x__2(t)+exp(t), diff(x__2(t),t) = x__1(t)-x__2(t)+exp(t), x__1(0) = 1, x__2(0) = 1], singsol=all)
\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-t^{2}-t +1\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-2 t^{2}+4\right )}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 31
DSolve[{x1'[t]==3*x1[t]-4*x2[t]+Exp[t],x2'[t]==1*x1[t]-1*x2[t]+Exp[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -e^t \left (t^2+t-1\right ) \\ \text {x2}(t)\to -\frac {1}{2} e^t \left (t^2-2\right ) \\ \end{align*}