78.5.15 problem 7.c

Internal problem ID [21054]
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.c
Date solved : Thursday, October 02, 2025 at 07:01:48 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y\\ y^{\prime }&=5 x \left (t \right )-2 y \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 36

ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = 5*x(t)-2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ y \left (t \right ) &= -c_1 \cos \left (t \right )+c_2 \sin \left (t \right )+2 c_1 \sin \left (t \right )+2 c_2 \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode={D[x[t],t]==2*x[t]-y[t],D[y[t],t]==5*x[t]-2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 (2 \sin (t)+\cos (t))-c_2 \sin (t)\\ y(t)&\to 5 c_1 \sin (t)+c_2 (\cos (t)-2 \sin (t)) \end{align*}

✓ Sympy. Time used: 0.051 (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(-5*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{5} - \frac {2 C_{2}}{5}\right ) \cos {\left (t \right )} - \left (\frac {2 C_{1}}{5} + \frac {C_{2}}{5}\right ) \sin {\left (t \right )}, \ y{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )}\right ] \]