85.86.5 problem 2 (a)

Internal problem ID [23022]
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 (a)
Date solved : Sunday, October 12, 2025 at 05:55:06 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ D\left (x \right )\left (0\right )&=10 \\ y \left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 21

ode:=[diff(diff(x(t),t),t) = -2*y(t), diff(y(t),t) = y(t)-diff(x(t),t)]; 
ic:=[x(0) = 0, D(x)(0) = 10, y(0) = 5]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 10-10 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 5 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 24

ode={D[x[t],{t,2}]==-2*y[t],D[y[t],t]==y[t]-D[x[t],t]}; 
ic={x[0]==0,Derivative[1][x][0] ==10,y[0]==5}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 10-10 e^{-t}\\ y(t)&\to 5 e^{-t} \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*y(t) + Derivative(x(t), (t, 2)),0),Eq(-y(t) + 2*Derivative(y(t), t),0)] 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 10): 4, y(0): 5} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 t \left (1 - 5 e^{5}\right ) + 40 e^{\frac {t}{2}} - 40, \ y{\left (t \right )} = 5 e^{\frac {t}{2}}\right ] \]