76.28.2 problem 3

Internal problem ID [17869]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:04:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+\sqrt {3}\, x_{2} \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \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: 1.066 (sec). Leaf size: 70

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 ) &= c_{2} \sinh \left (2 t \right )+\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 (-2 \sinh \left (2 t \right ) c_{1} +c_{2} \sinh \left (2 t \right )+\cosh \left (2 t \right ) c_{1} -2 c_{2} \cosh \left (2 t \right )+{\mathrm e}^{t}-2 \cosh \left (t \right )+2 \sinh \left (t \right )\right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.105 (sec). Leaf size: 127

DSolve[{D[x1[t],t]==1*x1[t]+Sqrt[3]*x2[t]+Exp[t],D[x2[t],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}{12} e^{-2 t} \left (-12 e^t-8 e^{3 t}+3 \left (3 c_1+\sqrt {3} c_2\right ) e^{4 t}+3 \left (c_1-\sqrt {3} c_2\right )\right ) \\ \text {x2}(t)\to \frac {1}{12} e^{-2 t} \left (8 \sqrt {3} e^t-4 \sqrt {3} e^{3 t}+3 \left (\sqrt {3} c_1+c_2\right ) e^{4 t}-3 \sqrt {3} c_1+9 c_2\right ) \\ \end{align*}