72.15.10 problem 19 (vii)

Internal problem ID [14812]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number : 19 (vii)
Date solved : Thursday, March 13, 2025 at 04:19:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 45

ode={D[x[t],t]==0*x[t]+1*y[t],D[y[t],t]==-4*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.098 (sec). Leaf size: 44

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

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{- 2 t} + \left (C_{1} + 2 C_{2}\right ) e^{- 2 t}, \ y{\left (t \right )} = - 4 C_{1} t e^{- 2 t} - 4 C_{2} e^{- 2 t}\right ] \]