85.86.6 problem 2 (b)

Internal problem ID [23023]
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 491
Problem number : 2 (b)
Date solved : Sunday, October 12, 2025 at 05:55:06 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ D\left (x \right )\left (0\right )&=1 \\ y \left (0\right )&=2 \\ D\left (y \right )\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 35

ode:=[diff(diff(y(t),t),t) = x(t)-2, diff(diff(x(t),t),t) = y(t)+2]; 
ic:=[x(0) = 0, D(x)(0) = 1, y(0) = 2, D(y)(0) = -3]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 38

ode={D[y[t],{t,2}]==x[t]-2,D[x[t],{t,2}]==y[t]+2}; 
ic={x[0]==0,Derivative[1][x][0] ==1,y[0]==2,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.212 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + Derivative(x(t), (t, 2)) + 2,0),Eq(-y(t) + Derivative(y(t), (t, 2)) - 2,0)] 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1, y(0): 2, Subs(Derivative(y(t), t), t, 0): -3} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {e^{t}}{2} + 2 - \frac {3 e^{- t}}{2}, \ y{\left (t \right )} = \frac {e^{t}}{2} - 2 + \frac {7 e^{- t}}{2}\right ] \]