14.18.2 problem 2

Internal problem ID [3872]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.6 (Variation of parameters for linear systems), page 624
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:58:20 AM
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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+4 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.156 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 74

ode={D[x1[t],t]==2*x1[t]-x2[t],D[x2[t],t]==-x1[t]+2*x2[t]+4*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 (4 t+c_1 \left (e^{2 t}+1\right )-c_2 e^{2 t}+2+c_2\right )\\ \text {x2}(t)&\to \frac {1}{2} e^t \left (4 t-c_1 e^{2 t}+c_2 e^{2 t}-2+c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 44

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

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