79.11.1 problem (c)

Internal problem ID [21109]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Differential Equations. Excercise XII at page 189
Problem number : (c)
Date solved : Thursday, October 02, 2025 at 07:08:31 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-4 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: 0.044 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 60

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

✓ Sympy. Time used: 0.039 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 4*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)
 

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