70.28.2 problem 3

Internal problem ID [19155]
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 : Thursday, October 02, 2025 at 03:38:44 PM
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*}

✓ Maple. Time used: 0.312 (sec). Leaf size: 70

ode:=[diff(x__1(t),t) = x__1(t)+3^(1/2)*x__2(t)+exp(t), diff(x__2(t),t) = 3^(1/2)*x__1(t)-x__2(t)+3^(1/2)*exp(-t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 127

ode={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]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) - sqrt(3)*x__2(t) - exp(t) + Derivative(x__1(t), t),0),Eq(-sqrt(3)*x__1(t) + x__2(t) + Derivative(x__2(t), t) - sqrt(3)*exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \sqrt {3} C_{1} e^{2 t} - \frac {\sqrt {3} C_{2} e^{- 2 t}}{3} - \frac {2 e^{t}}{3} - e^{- t}, \ x^{2}{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{- 2 t} - \frac {\sqrt {3} e^{t}}{3} + \frac {2 \sqrt {3} e^{- t}}{3}\right ] \]