74.22.10 problem 10

Internal problem ID [16586]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 10
Date solved : Monday, March 31, 2025 at 02:59:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.129 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 51

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

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

✓ Sympy. Time used: 0.107 (sec). Leaf size: 48

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

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