Plus in this direction.

Then, what would be this term look like?

So I have the force T plus dt, dx,

and dx and looking at it in positive direction.

So I put a plus over here.

And then the angle over here would be expressed

as dy until you see this expansion, dy,

dx plus d squared y,

dx squared dx simply meaning that

the incremental increment increase

of the angle would be this one.

I mean, the change of this

with respect to x therefore that is

dx squared as the length increased by dx.

That has to be balanced by equal Rho L dx

multiplied by d squared y, dt squared.

It looks complicated but,

physically meaning that the unbalanced force

in the Y direction has to be expressed as

mass times acceleration and apply

Newton's second law or upon the small element of a string

hoping that this will reveal

us the physical model

to understand the vibration of a string.

Now. That's complicated but,

let's look at some of the term.

This term, minus T, dy,

dx will be canceled out T multiplied dy,

dx and then we have T plus d squared y, dx squared dx.

So let's write down T d squared y,

dx squared dx,

and the other one is dt dx dx multiplied dy, dx.

So I have another term dt, dx, dy,

dx, and a dx,

and I have another term dt, dx,

dx and dx squared y, dx squared dx.

So I have dx squared.

As you remember, dx is small.

The squared of the small stuff will be much

much smaller than the small stuff of dx.

So I neglect it.

Then that has to be equal to Rho L d squared

y, dt squared dx.

Those dx are common.

Therefore, we got, we are approaching

to the final train station.

T, d squared y, dx squared,

plus dt, dx, dy.

Sorry, dy, dx.

That has to be equal to

Rho L. d squared y sorry, dt squared.