82.7.9 problem 34-12

Internal problem ID [21865]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-12
Date solved : Thursday, October 02, 2025 at 08:03:03 PM
CAS classification : system_of_ODEs

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

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

ode={D[x[t],t]==4*x[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 \frac {1}{4} e^{2 t} \left (2 c_1 \left (e^{4 t}+1\right )-c_2 \left (e^{4 t}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{2 t} \left (c_2 \left (e^{4 t}+1\right )-2 c_1 \left (e^{4 t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 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)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{2 t}}{2} - \frac {C_{2} e^{6 t}}{2}, \ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{6 t}\right ] \]