71.19.2 problem 2

Internal problem ID [14523]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 2
Date solved : Monday, March 31, 2025 at 12:29:29 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 28

ode:=[diff(x(t),t) = -x(t)+2*y(t), diff(y(t),t) = -2*x(t)+3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode={D[x[t],t]==-x[t]+2*y[t],D[y[t],t]==-2*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.091 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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