9.4.16 problem problem 16

Internal problem ID [980]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 16
Date solved : Tuesday, March 04, 2025 at 12:06:46 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-50 x_{1} \left (t \right )+20 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=100 x_{1} \left (t \right )-60 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -50*x__1(t)+20*x__2(t), diff(x__2(t),t) = 100*x__1(t)-60*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-100 t}+c_2 \,{\mathrm e}^{-10 t} \\ x_{2} \left (t \right ) &= -\frac {5 c_1 \,{\mathrm e}^{-100 t}}{2}+2 c_2 \,{\mathrm e}^{-10 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode={D[ x1[t],t]==-50*x1[t]+20*x2[t],D[ x2[t],t]==100*x1[t]-60*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-100 t} \left (c_1 \left (5 e^{90 t}+4\right )+2 c_2 \left (e^{90 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-100 t} \left (10 c_1 \left (e^{90 t}-1\right )+c_2 \left (4 e^{90 t}+5\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(50*x__1(t) - 20*x__2(t) + Derivative(x__1(t), t),0),Eq(-100*x__1(t) + 60*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {2 C_{1} e^{- 100 t}}{5} + \frac {C_{2} e^{- 10 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- 100 t} + C_{2} e^{- 10 t}\right ] \]