85.83.4 problem 1 (d)

Internal problem ID [23001]
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 444
Problem number : 1 (d)
Date solved : Sunday, October 12, 2025 at 05:55:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}y \left (t \right )&=x \left (t \right )\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 29

ode:=[diff(diff(y(t),t),t) = x(t), diff(diff(y(t),t),t) = y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 76

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

\begin{align*} x(t)&\to \frac {1}{54} e^{-t} \left (c_1 \left (27 e^{2 t}+1\right )+c_2 \left (27 e^{2 t}-1\right )\right )\\ y(t)&\to \frac {1}{54} e^{-t} \left (c_1 \left (27 e^{2 t}+1\right )+c_2 \left (27 e^{2 t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 44

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

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