# Linear First Order PDE- Looking for an example

• Feb 2nd 2011, 11:08 AM
bugatti79
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
• Feb 2nd 2011, 03:09 PM
Jester

$2u u_y = -1, \;\;u(x,0) = x^2?$
• Feb 2nd 2011, 10:23 PM
bugatti79
Quote:

Originally Posted by Danny

$2u u_y = -1, \;\;u(x,0) = x^2?$

Are we missing an x partial derivative?

As given above i get $u(y)=\sqrt{-y+c}$. Cant go any further?
• Feb 3rd 2011, 04:14 AM
Jester
It's a PDE so integrating gives (taking the positive solution)

$u(x,y) = \sqrt{f(x)-y}$.
• Feb 3rd 2011, 05:19 AM
bugatti79
Quote:

Originally Posted by Danny
It's a PDE so integrating gives (taking the positive solution)

$u(x,y) = \sqrt{f(x)-y}$.

Ok, the constant must be a function of x ie

$2\int{u} du=-\int{y} dy$ therefore

$u^2=-y+c$ but c=f(x)

$u(x,y)=+\sqrt{-y+f(x)}$ using boundary conditions give particular as

$u(x,y)=\sqrt{-y+x^4}$

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

Thanks
• Feb 3rd 2011, 08:51 AM
Jester
What if $y > x^4$?
• Feb 3rd 2011, 09:55 AM
bugatti79
Quote:

Originally Posted by Danny
What if $y > x^4$?

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 $-y+x^4 \not < 0$. Therefore $x^4 \geq y$. 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 $-y+x^4 \not < 0$.
Is there a better way of stating this mathematically?