63.18.7 problem 3(c)

Internal problem ID [13142]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 3(c)
Date solved : Monday, March 31, 2025 at 07:35:21 AM
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 )&=x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.118 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 29

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

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