44.29.8 problem 3(d)

Internal problem ID [9490]
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 A. Drill exercises. Page 400
Problem number : 3(d)
Date solved : Tuesday, September 30, 2025 at 06:19:24 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 )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 79

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

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

✓ Sympy. Time used: 0.093 (sec). Leaf size: 78

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) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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