85.88.12 problem 2 (b)

Internal problem ID [23039]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 499
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 09:17:42 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=7 \\ y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.067 (sec). Leaf size: 33

ode:=[diff(x(t),t)+2*y(t) = 4*exp(2*t), x(t)+diff(y(t),t)-y(t) = 2*exp(2*t)]; 
ic:=[x(0) = 7, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{-t}+3 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-t}-{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 37

ode={D[x[t],{t,1}]+2*y[t]==4*Exp[2*t], x[t]+ D[y[t],{t,1}]-y[t]==2*Exp[2*t]}; 
ic={x[0]==7,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} \left (3 e^{3 t}+4\right )\\ y(t)&\to -e^{-t} \left (e^{3 t}-2\right ) \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*y(t) - 4*exp(2*t) + Derivative(x(t), t),0),Eq(x(t) - y(t) - 2*exp(2*t) + Derivative(y(t), t),0)] 
ics = {x(0): 7, y(0): 1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 e^{2 t} + 4 e^{- t}, \ y{\left (t \right )} = - e^{2 t} + 2 e^{- t}\right ] \]