Please show your work Problem 10. Consider the second order differential equation d^2y/dt^2 + 6y =0 that represents the motion of a simple harmonic oscillator. Let y denote the velocity dy/dt. Write a system of differential equations in terms of the unknown functions y, u that is equivalent to the original second order differential equation. Also, find a value such that y(t) = sin (Beta t) is a solution of the second order differential equation. Solution y\"+6y=0 Let y\' =v Then y\" = v\' The equation becomes y\'+6integral y dt =0 ----------------------------------- When y\"+6y=0 the auxialary equation is m^2+6 =0 or m = 3i, -3i Hence solution is y = A sin 3t hence beta =3.

1. Please show your work Problem 10. Consider the second order differential equation d^2y/dt^2 +
6y =0 that represents the motion of a simple harmonic oscillator. Let y denote the velocity dy/dt.
Write a system of differential equations in terms of the unknown functions y, u that is equivalent
to the original second order differential equation. Also, find a value such that y(t) = sin (Beta t)
is a solution of the second order differential equation.
Solution
y"+6y=0
Let y' =v
Then y" = v'
The equation becomes
y'+6integral y dt =0
-----------------------------------
When y"+6y=0 the auxialary equation is
m^2+6 =0 or m = 3i, -3i
Hence solution is
y = A sin 3t hence beta =3