78.28.3 problem 1 (c)

Internal problem ID [18394]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 56. Homogeneous Linear Systems with Constant Coefficients. Problems at page 505
Problem number : 1 (c)
Date solved : Thursday, March 13, 2025 at 11:55:31 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right ) \\ y &= -\frac {{\mathrm e}^{3 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 46

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

\begin{align*} x(t)\to e^{3 t} (2 c_1 t+4 c_2 t+c_1) \\ y(t)\to e^{3 t} (c_2-(c_1+2 c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 41

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

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