72.9.3 problem 3

Internal problem ID [14720]
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 3.1. page 258
Problem number : 3
Date solved : Thursday, March 13, 2025 at 04:16:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 20

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

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y &= \left (2 c_{2} t +c_{1} \right ) {\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 26

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

\begin{align*} x(t)\to c_1 e^t \\ y(t)\to e^t (2 c_1 t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 24

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

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