10.17.8 problem 8

Internal problem ID [1423]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 8
Date solved : Saturday, March 29, 2025 at 10:54:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+\frac {3 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {3 x_{1} \left (t \right )}{2}+\frac {x_{2} \left (t \right )}{2} \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 3\\ x_{2} \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 28

ode:=[diff(x__1(t),t) = -5/2*x__1(t)+3/2*x__2(t), diff(x__2(t),t) = -3/2*x__1(t)+1/2*x__2(t)]; 
ic:=x__1(0) = 3x__2(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (-6 t +3\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-18 t -3\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 31

ode={D[ x1[t],t]==-5/2*x1[t]+3/2*x2[t],D[ x2[t],t]==-3/2*x1[t]+1/2*x2[t]}; 
ic={x1[0]==3,x2[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} (3-6 t) \\ \text {x2}(t)\to -e^{-t} (6 t+1) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(5*x__1(t)/2 - 3*x__2(t)/2 + Derivative(x__1(t), t),0),Eq(3*x__1(t)/2 - x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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