78.5.20 problem 7.h

Internal problem ID [21059]
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.h
Date solved : Thursday, October 02, 2025 at 07:01:51 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y\\ y^{\prime }&=2 x \left (t \right )-3 y \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 34

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: 46

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-c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.050 (sec). Leaf size: 36

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)
 

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