28.4.21 problem 7.21

Internal problem ID [4553]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.21
Date solved : Tuesday, March 04, 2025 at 06:52:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-2 x \left (t \right )+y \left (t \right )&=5 \,{\mathrm e}^{t} \cos \left (t \right )\\ x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=10 \,{\mathrm e}^{t} \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.145 (sec). Leaf size: 35

ode:=[diff(x(t),t)-2*x(t)+y(t) = 5*exp(t)*cos(t), x(t)+diff(y(t),t)-2*y(t) = 10*exp(t)*sin(t)]; 
ic:=x(0) = 0y(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x &= -5 \,{\mathrm e}^{t} \cos \left (t \right )+5 \sin \left (t \right ) {\mathrm e}^{t}+5 \,{\mathrm e}^{t} \\ y &= -5 \,{\mathrm e}^{t} \cos \left (t \right )+5 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 30

ode={D[x[t],t]-2*x[t]+y[t]==5*Exp[t]*Cos[t],x[t]+D[y[t],t]-2*y[t]==10*Exp[t]*Sin[t]}; 
ic={x[0]==0,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -5 e^t (-\sin (t)+\cos (t)-1) \\ y(t)\to -5 e^t (\cos (t)-1) \\ \end{align*}

✓ Sympy. Time used: 0.288 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + y(t) - 5*exp(t)*cos(t) + Derivative(x(t), t),0),Eq(x(t) - 2*y(t) - 10*exp(t)*sin(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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