20.13.13 problem 13

Internal problem ID [3822]
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 : 13
Date solved : Tuesday, March 04, 2025 at 05:17:34 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 )+t\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+1 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 44

ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t)+t, diff(x__2(t),t) = -2*x__1(t)+x__2(t)+1]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 79

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

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

✓ Sympy. Time used: 0.133 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-t + 2*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - x__2(t) + Derivative(x__2(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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