Internal problem ID [779]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number: 2.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )+\sqrt {3}\, x_{2} \relax (t )+{\mathrm e}^{t}\\ x_{2}^{\prime }\relax (t )&=\sqrt {3}\, x_{1} \relax (t )-x_{2} \relax (t )+\sqrt {3}\, {\mathrm e}^{-t} \end {align*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 67
dsolve([diff(x__1(t),t)=1*x__1(t)+sqrt(3)*x__2(t)+exp(t),diff(x__2(t),t)=sqrt(3)*x__1(t)-1*x__2(t)+sqrt(3)*exp(-t)],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = {\mathrm e}^{2 t} \sqrt {3}\, c_{1}-\frac {\sqrt {3}\, {\mathrm e}^{-2 t} c_{2}}{3}-\frac {2 \,{\mathrm e}^{t}}{3}-{\mathrm e}^{-t} \] \[ x_{2} \relax (t ) = {\mathrm e}^{-2 t} c_{2}+c_{1} {\mathrm e}^{2 t}+\frac {2 \sqrt {3}\, {\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{t} \sqrt {3}}{3} \]
✓ Solution by Mathematica
Time used: 2.487 (sec). Leaf size: 240
DSolve[{x1'[t]==1*x1[t]+Sqrt[4]*x2[t]+Exp[t],x2'[t]==Sqrt[3]*x1[t]-1*x2[t]+Sqrt[3]*Exp[-t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -\frac {e^t}{\sqrt {3}}+\sinh (t)-\cosh (t)+c_1 \cosh \left (\sqrt {1+2 \sqrt {3}} t\right )+\frac {(c_1+2 c_2) \sinh \left (\sqrt {1+2 \sqrt {3}} t\right )}{\sqrt {1+2 \sqrt {3}}} \\ \text {x2}(t)\to \frac {1}{4} \left (4 e^{-t}-2 e^t+\frac {2 \left (\left (6+\sqrt {3}\right ) c_1+\left (1+2 \sqrt {3}\right ) \left (\sqrt {1+2 \sqrt {3}}-1\right ) c_2\right ) e^{\sqrt {1+2 \sqrt {3}} t}}{\left (1+2 \sqrt {3}\right )^{3/2}}+\frac {\left (2 \left (1+2 \sqrt {3}\right ) \left (1+\sqrt {1+2 \sqrt {3}}\right ) c_2-2 \left (6+\sqrt {3}\right ) c_1\right ) e^{-\sqrt {1+2 \sqrt {3}} t}}{\left (1+2 \sqrt {3}\right )^{3/2}}\right ) \\ \end{align*}