85.89.1 problem 5

Internal problem ID [23045]
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. B Exercises at page 499
Problem number : 5
Date solved : Sunday, October 12, 2025 at 05:55:07 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 51

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t}+c_1 \cos \left (t \right )+c_2 \,{\mathrm e}^{t}+c_3 \sin \left (t \right )+c_4 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= -c_1 \cos \left (t \right )+c_2 \,{\mathrm e}^{t}-c_3 \sin \left (t \right )+c_4 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 157

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

\begin{align*} x(t)&\to \frac {1}{4} \left (e^{-2 t} \left ((c_1-c_2+c_3-c_4) e^t+(c_1+c_2+c_3+c_4) e^{3 t}+4\right )+2 (c_1-c_3) \cos (t)+2 (c_2-c_4) \sin (t)\right )\\ y(t)&\to \frac {1}{4} e^{-t} \left (c_1 e^{2 t}+c_2 e^{2 t}+c_3 e^{2 t}+c_4 e^{2 t}-2 (c_1-c_3) e^t \cos (t)-2 (c_2-c_4) e^t \sin (t)+c_1-c_2+c_3-c_4\right ) \end{align*}

✓ Sympy. Time used: 0.300 (sec). Leaf size: 109

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

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