64.24.4 problem 4

Internal problem ID [13610]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Nonlinear differential equations. Section 13.2, Exercises page 656
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 10:05:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -{\mathrm e}^{-3 t} c_{1} +\frac {c_{2} {\mathrm e}^{3 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

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

\begin{align*} x(t)\to \frac {1}{6} e^{-3 t} \left (c_1 \left (5 e^{6 t}+1\right )+5 c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-3 t} \left (c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+5\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - 5*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 3 t} + 5 C_{2} e^{3 t}, \ y{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{3 t}\right ] \]