68.1.60 problem 82

Internal problem ID [17127]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 82
Date solved : Thursday, October 02, 2025 at 01:44:08 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 19

ode:=[diff(x(t),t) = 4*y(t), diff(y(t),t) = -4*x(t)]; 
ic:=[x(0) = 4, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 \cos \left (4 t \right ) \\ y \left (t \right ) &= -4 \sin \left (4 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 20

ode={D[x[t],t]==4*y[t],D[y[t],t]==-4*x[t]}; 
ic={x[0]==4,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 4 \cos (4 t)\\ y(t)&\to -4 \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.037 (sec). Leaf size: 31

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

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