76.8.11 problem 11

Internal problem ID [17421]
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 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 11
Date solved : Thursday, March 13, 2025 at 10:08:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {3 x \left (t \right )}{4}-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-\frac {5 y \left (t \right )}{4} \end{align*}

✓ Maple. Time used: 0.074 (sec). Leaf size: 45

ode:=[diff(x(t),t) = 3/4*x(t)-2*y(t), diff(y(t),t) = x(t)-5/4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{4}} \left (c_{1} \sin \left (t \right )+c_{2} \sin \left (t \right )-\cos \left (t \right ) c_{1} +c_{2} \cos \left (t \right )\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 56

ode={D[x[t],t]==3/4*x[t]-2*y[t],D[y[t],t]==x[t]-5/4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t/4} (c_1 \cos (t)+(c_1-2 c_2) \sin (t)) \\ y(t)\to e^{-t/4} (c_2 \cos (t)+(c_1-c_2) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t)/4 + 2*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 5*y(t)/4 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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