14.13.14 problem 14

Internal problem ID [3823]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 06:57:47 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 )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )+5 \,{\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.166 (sec). Leaf size: 52

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+exp(2*t), diff(x__2(t),t) = 3*x__1(t)-x__2(t)+5*exp(2*t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 76

ode={D[x1[t],t]==x1[t]+x2[t]+Exp[2*t],D[x2[t],t]==3*x1[t]-x2[t]+5*Exp[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{4} e^{-2 t} \left (e^{4 t} (8 t-1+3 c_1+c_2)+c_1-c_2\right )\\ \text {x2}(t)&\to \frac {1}{4} e^{-2 t} \left (e^{4 t} (8 t+3+3 c_1+c_2)-3 c_1+3 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) - x__2(t) - exp(2*t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + x__2(t) - 5*exp(2*t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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