82.6.5 problem 33-38

Internal problem ID [21852]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 33. Systems of ordinary differential equations. Page 1059
Problem number : 33-38
Date solved : Thursday, October 02, 2025 at 08:02:55 PM
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: 0.040 (sec). Leaf size: 31

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 ) &= {\mathrm e}^{2 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= {\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]==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} (-4 c_1 t+2 c_2 t+c_2) \end{align*}

✓ Sympy. Time used: 0.059 (sec). Leaf size: 46

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)
 

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