Question

(From 4.2) In a mass-spring system, motion is assumed to occur only in the vertical direction. That is, the system has one degree of freedom. When the mass is pulled downward and then released, the system will oscillate. If the system is undamped, meaning that there are no forces to slow or stop the oscillation, then the system will oscillate indefinitely. Applying Newton’s Second Law of Motion to the mass yield the second-order differential equation ? ′′ + ? 2? = 0 where ? is the displacement at time ? and ? is a fixed constant called the natural frequency of the system.

a) Verify that the general solution of the above differential equation is ?(?) = ?1????? + ?2????? where ?1 and ?2 are arbitrary constants.

b) Show that the set of all functions ?(?) form a vector space.

Answer 1

a) Differentiate twice with respect to t gives

Hence the function is a solution to the given equation

b) For    we get ,

So the set contains the zero element.

As, are scalars, So and any comination of function of the form , is a solution of

So the set is closed.

Adding of functions is associative and zero element exists

Also inverse of any function of the form is

Also its obvious that the sum if abelian.

Again is a solution of , so is

Hence the multiplication distributive as well.

Hence all the conditions are satisfied

So the set is a vector space.