87.12.33 problem 36

Internal problem ID [23481]
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 2. Linear differential equations. Exercise at page 93
Problem number : 36
Date solved : Sunday, October 12, 2025 at 05:55:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}y \left (t \right )+6 x \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )-\frac {d}{d t}x \left (t \right )+6 y \left (t \right )&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 62

ode:=[diff(diff(x(t),t),t)+diff(y(t),t)+6*x(t) = 0, diff(diff(y(t),t),t)-diff(x(t),t)+6*y(t) = 0]; 
ic:=[D(x)(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (3 t \right )+c_2 \cos \left (3 t \right )-\frac {3 c_1 \sin \left (2 t \right )}{2}+\frac {3 c_2 \cos \left (2 t \right )}{2} \\ y \left (t \right ) &= -c_1 \cos \left (3 t \right )+c_2 \sin \left (3 t \right )-\frac {3 c_1 \cos \left (2 t \right )}{2}-\frac {3 c_2 \sin \left (2 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 82

ode={D[x[t],{t,2}]+D[y[t],t]+6*x[t]==0,D[y[t],{t,2}]-D[x[t],t]+6*y[t]==0}; 
ic={Derivative[1][x][0] ==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{5} (3 c_1 \cos (2 t)+2 c_1 \cos (3 t)+c_3 (3 \sin (2 t)-2 \sin (3 t)))\\ y(t)&\to \frac {1}{5} (3 c_3 \cos (2 t)+2 c_3 \cos (3 t)+c_1 (2 \sin (3 t)-3 \sin (2 t))) \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 71

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*x(t) + Derivative(x(t), (t, 2)) + Derivative(y(t), t),0),Eq(6*y(t) - Derivative(x(t), t) + Derivative(y(t), (t, 2)),0)] 
ics = {Subs(Derivative(x(t), t), t, 0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{2} \sin {\left (2 t \right )}}{2} - \frac {C_{2} \sin {\left (3 t \right )}}{3} + \frac {C_{4} \cos {\left (2 t \right )}}{2} + \frac {C_{4} \cos {\left (3 t \right )}}{3}, \ y{\left (t \right )} = \frac {C_{2} \cos {\left (2 t \right )}}{2} + \frac {C_{2} \cos {\left (3 t \right )}}{3} - \frac {C_{4} \sin {\left (2 t \right )}}{2} + \frac {C_{4} \sin {\left (3 t \right )}}{3}\right ] \]