76.9.8 problem 8

Internal problem ID [17447]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 8
Date solved : Monday, March 31, 2025 at 04:13:59 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.143 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 31

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

\begin{align*} x(t)\to e^{-t} (3-6 t) \\ y(t)\to -e^{-t} (6 t+1) \\ \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 46

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

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