87.29.15 problem 15

Internal problem ID [23895]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 15
Date solved : Thursday, October 02, 2025 at 09:46:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 26

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = -x__1(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= c_1 \cos \left (t \right )-c_2 \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 31

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-x1[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to c_1 \cos (t)+c_2 \sin (t)\\ \text {x2}(t)&\to c_2 \cos (t)-c_1 \sin (t) \end{align*}

✓ Sympy. Time used: 0.034 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(x1(t) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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