10.17.3 problem 3

Internal problem ID [1418]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 3
Date solved : Saturday, March 29, 2025 at 10:54:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.121 (sec). Leaf size: 31

ode:=[diff(x__1(t),t) = -3/2*x__1(t)+x__2(t), diff(x__2(t),t) = -1/4*x__1(t)-1/2*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 54

ode={D[ x1[t],t]==-3/2*x1[t]+1*x2[t],D[ x2[t],t]==-1/4*x1[t]-1/2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.093 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t)/2 - x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t)/4 + x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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