7.23.9 problem 9

Internal problem ID [595]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 11:27:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 54

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

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+{\mathrm e}^{t} c_1 -\frac {7 \cos \left (2 t \right )}{5}-\frac {4 \sin \left (2 t \right )}{5} \\ y \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{t} c_1}{3}-\frac {4 \sin \left (2 t \right )}{5}-\frac {2 \cos \left (2 t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 102

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

\begin{align*} x(t)\to \frac {1}{10} \left (-8 \sin (2 t)-14 \cos (2 t)+5 e^{-t} \left (c_1 \left (3 e^{2 t}-1\right )-3 c_2 \left (e^{2 t}-1\right )\right )\right ) \\ y(t)\to \frac {1}{10} \left (-8 \sin (2 t)-4 \cos (2 t)+e^{-t} \left (5 c_1 \left (e^{2 t}-1\right )-5 c_2 \left (e^{2 t}-3\right )\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.298 (sec). Leaf size: 60

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

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} + 3 C_{2} e^{t} - \frac {4 \sin {\left (2 t \right )}}{5} - \frac {7 \cos {\left (2 t \right )}}{5}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} - \frac {4 \sin {\left (2 t \right )}}{5} - \frac {2 \cos {\left (2 t \right )}}{5}\right ] \]