76.9.4 problem 4

Internal problem ID [17435]
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 : 4
Date solved : Thursday, March 13, 2025 at 10:08:19 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (5 c_{2} t +5 c_{1} +2 c_{2} \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 59

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

\begin{align*} x(t)\to \frac {1}{2} e^{-t/2} (c_1 (2-5 t)+5 c_2 t) \\ y(t)\to \frac {1}{2} e^{-t/2} (-5 c_1 t+5 c_2 t+2 c_2) \\ \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 53

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

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