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 }\left (t \right )&=x_{1} \left (t \right )+\sqrt {3}\, x_{2} \left (t \right )+{\mathrm e}^{t}\\ x_{2}^{\prime }\left (t \right )&=\sqrt {3}\, x_{1} \left (t \right )-x_{2} \left (t \right )+\sqrt {3}\, {\mathrm e}^{-t} \end {align*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 71
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)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= \sinh \left (2 t \right ) c_{2} +\cosh \left (2 t \right ) c_{1} -\frac {5 \cosh \left (t \right )}{3}+\frac {\sinh \left (t \right )}{3} \\ x_{2} \left (t \right ) &= -\frac {\sqrt {3}\, \left (\cosh \left (2 t \right ) c_{1} -2 \cosh \left (2 t \right ) c_{2} -2 \sinh \left (2 t \right ) c_{1} +\sinh \left (2 t \right ) c_{2} +{\mathrm e}^{t}+2 \sinh \left (t \right )-2 \cosh \left (t \right )\right )}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.501 (sec). Leaf size: 313
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 {1}{6} \left (-6 e^{-t}-\frac {2 \left (6+\sqrt {3}\right ) e^t}{1+2 \sqrt {3}}+\frac {\left (3 \left (\sqrt {1+2 \sqrt {3}}-1\right ) c_1-6 c_2\right ) e^{-\sqrt {1+2 \sqrt {3}} t}}{\sqrt {1+2 \sqrt {3}}}+\frac {3 \left (\left (1+\sqrt {1+2 \sqrt {3}}\right ) c_1+2 c_2\right ) e^{\sqrt {1+2 \sqrt {3}} t}}{\sqrt {1+2 \sqrt {3}}}\right ) \\ \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*}