10.18.8 problem 8

Internal problem ID [1435]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 12:35:51 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 44

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

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{t}+{\mathrm e}^{-t} c_1 +2 t \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t} c_1 -{\mathrm e}^{t}+2 t \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 80

ode={D[ x1[t],t]==2*x1[t]-1*x2[t]+Exp[t],D[ x2[t],t]==3*x1[t]-2*x2[t]-Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.146 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) + x__2(t) - exp(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + 2*x__2(t) + exp(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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