7.23.8 problem 8

Internal problem ID [594]
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 : 8
Date solved : Saturday, March 29, 2025 at 04:57:31 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 )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right )-{\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.183 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} c_2 +{\mathrm e}^{3 t} c_1 +{\mathrm e}^{2 t} \\ y \left (t \right ) &= -{\mathrm e}^{t} c_2 +{\mathrm e}^{3 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 66

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

\begin{align*} x(t)\to \frac {1}{2} e^t \left (2 e^t+(c_1+c_2) e^{2 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^t \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 32

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(-x(t) - 2*y(t) + exp(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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