44.28.8 problem 1(h)

Internal problem ID [9481]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section 10.3 Homogeneous Linear Systems with Constant Coefficients. Page 387
Problem number : 1(h)
Date solved : Tuesday, September 30, 2025 at 06:19:19 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.141 (sec). Leaf size: 57

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 0.070 (sec). Leaf size: 63

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

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