20.20.1 problem 1

Internal problem ID [3891]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 1
Date solved : Friday, March 14, 2025 at 01:27:30 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 29

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

\begin{align*} \text {x1}(t)\to c_1 e^{(t-1) t} \\ \text {x2}(t)\to c_2 e^t-c_1 \\ \end{align*}

✓ Sympy. Time used: 1.360 (sec). Leaf size: 19

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

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