66.10.4 problem 4

Internal problem ID [16102]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.2. page 277
Problem number : 4
Date solved : Thursday, October 02, 2025 at 10:42:30 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )+4 y \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= {\mathrm e}^{3 t} \left (c_2 t +c_1 +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 44

ode={D[x[t],t]==2*x[t]+1*y[t],D[y[t],t]==-x[t]+4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{3 t} (c_1 (-t)+c_2 t+c_1)\\ y(t)&\to e^{3 t} ((c_2-c_1) t+c_2) \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - y(t) + Derivative(x(t), t),0),Eq(x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{2} t e^{3 t} - \left (C_{1} - C_{2}\right ) e^{3 t}, \ y{\left (t \right )} = - C_{1} e^{3 t} - C_{2} t e^{3 t}\right ] \]