As user, we first needed to solve it's corresponding

homogeneous problem say, y double prime + 4y = 0.

Its characteristic equation is r squared + 4 =0.

So that r is equal to + or- 2i, okay?

So r = + or- 2i and that means

the complementary solution will be y of c,

which is the c1 cosine 2x + c2 sine of 2x, right?

Now look at the right hand side, okay?

-4 times the sine squared x, right?

By the trigonometric identity sine squared x

= one-half times 1- cosine 2x, okay?

So we can find the annihilator of sine squared x,

as D times D squared + 4, sine squared x.

In other words D times (D squared + 4),

(1- cosine 2x) will be equal to 0, okay?

So, D times (D squared + 4) is annihilator of this

right-hand side, -4 sine squared x, right?

And what I mean is D( D squared + 4)

acting on -4 sine squared x and

that is = 0, right?

That an annihilator, okay?

We found it, okay?

Apply this differential operator to the given one,

then you're going to get in fact the following, D and D squared + 4.

Now you have here one more D squared + 4 and y and that = 0, okay?

So in fact, this is D(D squared + 4) squared and y = 0, right?

That's the equation we get in high order

homogeneous differential equation.

You can write down the general solution of this high order homogeneous

constant coefficient differential equation easily, okay?

Thus the general solution would be y = first

the complementary solution yc down here plus you

need three more linear independent solutions.

One is coming from D, that is equal to 1,

another one is coming from the (D squared + 4) squared so

they are x times cosine 2x and x times the sine of 2x, okay?

So the general solution of this high order

homogeneous differential equation

is yc + yc + d1 + d2x cosine 2x + d3x sine 2x.

And that means what?

That means there must be a particular solution of

this given the differential equation in the form

of d1 + d2x cosine 2x + d3 x sine 2x, right?