78.5.13 problem 7.a

Internal problem ID [21052]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.a
Date solved : Thursday, October 02, 2025 at 07:01:47 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.161 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

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

✓ Sympy. Time used: 0.035 (sec). Leaf size: 22

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) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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