76.8.4 problem 4

Internal problem ID [17422]
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 : 4
Date solved : Monday, March 31, 2025 at 04:13:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-\frac {5 y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=\frac {9 x \left (t \right )}{5}-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 57

ode:=[diff(x(t),t) = 2*x(t)-5/2*y(t), diff(y(t),t) = 9/5*x(t)-y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_1 +\cos \left (\frac {3 t}{2}\right ) c_2 \right ) \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_1 +\sin \left (\frac {3 t}{2}\right ) c_2 -\cos \left (\frac {3 t}{2}\right ) c_1 +\cos \left (\frac {3 t}{2}\right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 84

ode={D[x[t],t]==2*x[t]-5/2*y[t],D[y[t],t]==9/5*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{t/2} \left (3 c_1 \cos \left (\frac {3 t}{2}\right )+(3 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ y(t)\to \frac {1}{5} e^{t/2} \left (5 c_2 \cos \left (\frac {3 t}{2}\right )+(6 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 5*y(t)/2 + Derivative(x(t), t),0),Eq(-9*x(t)/5 + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {5 C_{1}}{6} - \frac {5 C_{2}}{6}\right ) e^{\frac {t}{2}} \cos {\left (\frac {3 t}{2} \right )} - \left (\frac {5 C_{1}}{6} + \frac {5 C_{2}}{6}\right ) e^{\frac {t}{2}} \sin {\left (\frac {3 t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {t}{2}} \cos {\left (\frac {3 t}{2} \right )} - C_{2} e^{\frac {t}{2}} \sin {\left (\frac {3 t}{2} \right )}\right ] \]