76.28.4 problem 5

Internal problem ID [17792]
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 : 5
Date solved : Thursday, March 13, 2025 at 10:49:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+{\mathrm e}^{-2 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 44

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+exp(-2*t), diff(x__2(t),t) = 4*x__1(t)-2*x__2(t)-2*exp(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} c_{2}}{4}+c_{1} {\mathrm e}^{2 t}+\frac {{\mathrm e}^{t}}{2} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-3 t} c_{2} +c_{1} {\mathrm e}^{2 t}-{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.615 (sec). Leaf size: 84

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+Exp[-2*t],D[x2[t],t]==4*x1[t]-2*x2[t]-2*Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {e^t}{2}+\frac {1}{5} (c_1-c_2) e^{-3 t}+\frac {1}{5} (4 c_1+c_2) e^{2 t} \\ \text {x2}(t)\to \frac {1}{5} e^{-3 t} \left (-5 e^t+(4 c_1+c_2) e^{5 t}-4 c_1+4 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.213 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) - x__2(t) + Derivative(x__1(t), t) - exp(-2*t),0),Eq(-4*x__1(t) + 2*x__2(t) + 2*exp(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- 3 t}}{4} + C_{2} e^{2 t} + \frac {e^{t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{2 t} - e^{- 2 t}\right ] \]