76.10.9 problem 9

Internal problem ID [17460]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 9
Date solved : Monday, March 31, 2025 at 04:14:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.134 (sec). Leaf size: 49

ode:=[diff(x(t),t) = 2*x(t)-4*y(t), diff(y(t),t) = 2*x(t)-2*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 48

ode={D[x[t],t]==2*x[t]-4*y[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 \cos (2 t)+(c_1-2 c_2) \sin (2 t) \\ y(t)\to c_2 \cos (2 t)+(c_1-c_2) \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 4*y(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)
 

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