# Thread: Linear First Order PDE- Looking for an example

1. ## 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

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

3. Originally Posted by Danny

$\displaystyle 2u u_y = -1, \;\;u(x,0) = x^2?$
Are we missing an x partial derivative?

As given above i get $\displaystyle u(y)=\sqrt{-y+c}$. Cant go any further?

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

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

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

$\displaystyle u(x,y) = \sqrt{f(x)-y}$.
Ok, the constant must be a function of x ie

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

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

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

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

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

Thanks

6. What if $\displaystyle y > x^4$?

7. Originally Posted by Danny
What if $\displaystyle y > x^4$?
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 $\displaystyle -y+x^4 \not < 0$. Therefore $\displaystyle x^4 \geq y$. 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 $\displaystyle -y+x^4 \not < 0$.
Is there a better way of stating this mathematically?