Hi, in this video we will talk about how to solve this differential equation exactly.

In the previous video there was a mistake on the project sheet.

It said that U of 0 and U of 1 were 0 which is obviously wrong.

So the temperature P has boundary values not the thermal forcing U.

So P is the temperature and U is the heat source or sig.

It's just a matter of sine.

Okay, so how do we solve this?

In one dimension we can do this.

So the easiest thing to do is just integrate of course.

So the question here is what is P of x depending obviously on U.

And what do we do?

We integrate once.

So this is actually just minus P prime of x nothing else.

So what happens if we integrate minus integral 0 to x of P prime of y dy is equal to the

integral from 0 to x of U of y dy.

And this is the same thing as minus P prime of x plus P prime of 0 is equal to this integral

here.

So 0 dy and well let's maybe change the x to z so I don't have to change variables again.

So if we first integrate up to z and then we integrate another time.

So minus integral P prime of z from 0 to x plus P prime of 0 dz is equal to another integral

0 to x 0 to z U of y dy dz.

So this is a double integral.

What's the first term?

That is minus P of x plus P of 0 which is equal to 0 plus x times P prime of 0.

This is equal to well this doesn't change at all 0 to x 0 to z U of y dy dz.

Okay so what does that mean?

This means that P of x is equal to x times P prime of 0 minus integral 0 x 0 z U of y

dy dz.

And we're almost done but not yet because we don't know what P prime of 0 is.

One thing we haven't used before is the boundary condition P of 1 is equal to 0.

So this we have to somehow use in order to fix P prime of 0.

We know that P of 1 is equal to 0.

This means that 0 is equal to well P of 1 which is 1 times P prime of 0 minus the integral

0 1 0 z U of y dy dz.

So this means that P prime of 0 is equal to this double integral.

This means plugging in P of x is equal to x times this double integral

minus this double integral.

So we could write this by combining those two integrals but doesn't really matter actually.

And as you can see for x equal to 0 this part is 0 and this part is 0 so P of 0 is still

0 and P of 1 is still 1.

These two parts which are subtracted from each other are equal which means that the

whole term is then zero.

This term fulfills the boundary conditions and if you were to differentiate P of x twice

again and not then you would get back minus U of x which is exactly what we started with.

this is the explicit formula for P as expressed in terms of U and that is what

we will have to discretize then.