79.12.2 problem (b)

Internal problem ID [21111]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Differential Equations. Excercise XIII at page 189
Problem number : (b)
Date solved : Thursday, October 02, 2025 at 07:08:32 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=8 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: 0.050 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 8*x(t)+y(t), diff(y(t),t) = -4*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{6 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= -{\mathrm e}^{6 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]==8*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 e^{6 t} (2 c_1 t+c_2 t+c_1)\\ y(t)&\to e^{6 t} (c_2-2 (2 c_1+c_2) t) \end{align*}

✓ Sympy. Time used: 0.059 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-8*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)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{6 t} + \left (C_{1} + 2 C_{2}\right ) e^{6 t}, \ y{\left (t \right )} = - 4 C_{1} t e^{6 t} - 4 C_{2} e^{6 t}\right ] \]