14.18.1 problem 1

Internal problem ID [3871]
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 : 1
Date solved : Tuesday, September 30, 2025 at 06:58:19 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.158 (sec). Leaf size: 65

ode:=[diff(x__1(t),t) = 4*x__1(t)-3*x__2(t)+exp(2*t), diff(x__2(t),t) = 2*x__1(t)-x__2(t)+exp(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \left (c_2 +3 t \right ) {\mathrm e}^{t}+{\mathrm e}^{2 t} c_1 +3 \,{\mathrm e}^{2 t} t -3 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= \left (c_2 +3 t -1\right ) {\mathrm e}^{t}+\frac {2 \,{\mathrm e}^{2 t} c_1}{3}+2 \,{\mathrm e}^{2 t} t -\frac {8 \,{\mathrm e}^{2 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 73

ode={D[x1[t],t]==4*x1[t]-3*x2[t]+Exp[2*t],D[x2[t],t]==2*x1[t]-x2[t]+Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.105 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + 3*x__2(t) - exp(2*t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + x__2(t) - exp(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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