14.18.3 problem 3

Internal problem ID [3873]
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 : 3
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 )&=3 x_{1} \left (t \right )+x_{2} \left (t \right )+t \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{2} \left (t \right )+{\mathrm e}^{3 t} \end{align*}

✓ Maple. Time used: 0.196 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 0.073 (sec). Leaf size: 41

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

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