85.86.1 problem 1 (a)

Internal problem ID [23018]
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 : 1 (a)
Date solved : Thursday, October 02, 2025 at 09:17:33 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.055 (sec). Leaf size: 22

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

\[ \left [ x{\left (t \right )} = - \sin {\left (t \right )} + 2 \cos {\left (t \right )}, \ y{\left (t \right )} = - 2 \sin {\left (t \right )} - \cos {\left (t \right )}\right ] \]