75.27.1 problem 776

Internal problem ID [17158]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 776
Date solved : Thursday, March 13, 2025 at 09:18:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-9 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 35

ode:=[diff(x(t),t) = -9*y(t), diff(y(t),t) = x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= -\frac {\cos \left (3 t \right ) c_{1}}{3}+\frac {\sin \left (3 t \right ) c_{2}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 42

ode={D[x[t],t]==-9*y[t],D[y[t],t]==x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 \cos (3 t)-3 c_2 \sin (3 t) \\ y(t)\to c_2 \cos (3 t)+\frac {1}{3} c_1 \sin (3 t) \\ \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(9*y(t) + Derivative(x(t), t),0),Eq(-x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{1} \sin {\left (3 t \right )} - 3 C_{2} \cos {\left (3 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )}\right ] \]