Thread: Linear First Order PDE- Looking for an example

Hi All,

I have an examinable question that I am stuck on and obviously I cannot submit it here.
Instead I am asking if some one could provide a simple problem that demonstrates the principle. Here goes:

We have linear first order PDE u(x,y). Asked to find u given intial conditions which I can.
Next it asks to show that the solution is not defined when y > f(x). (I know what the f(x) of x is but cannot show it)

Could anyone provide a simple example to demonstrate this or at least what do i do?

Thanks
bugatti79

How about something like

Originally Posted by Danny

How about something like

Are we missing an x partial derivative?

As given above i get . Cant go any further?

It's a PDE so integrating gives (taking the positive solution)

.

Originally Posted by Danny

It's a PDE so integrating gives (taking the positive solution)

.

Ok, the constant must be a function of x ie

therefore

but c=f(x)

using boundary conditions give particular as



How do we use this to show the solution is not defined when y> some function of x?

Thanks

What if ?

Originally Posted by Danny

What if ?

Ok, after some reading and with your help my interpretation is the following:

there are 2 restrictions: We never divide by 0 and we assume real values functions. Therefore based on this and looking at the above example, the quantity . Therefore . Otherwise we get a complex number.

What is the domain of the function?

The domain of this function is the set of all real values of x and y whose quantity .
Is there a better way of stating this mathematically?