10.17.2 problem 2

Internal problem ID [1417]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 2
Date solved : Saturday, March 29, 2025 at 10:54:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = 4*x__1(t)-2*x__2(t), diff(x__2(t),t) = 8*x__1(t)-4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 t +c_2 \\ x_{2} \left (t \right ) &= -\frac {1}{2} c_1 +2 c_1 t +2 c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

ode={D[ x1[t],t]==4*x1[t]-2*x2[t],D[ x2[t],t]==8*x1[t]-4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 4 c_1 t-2 c_2 t+c_1 \\ \text {x2}(t)\to 8 c_1 t-4 c_2 t+c_2 \\ \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) + 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 4 C_{1} t + C_{1} + 4 C_{2}, \ x^{2}{\left (t \right )} = 8 C_{1} t + 8 C_{2}\right ] \]