How to find uniqueness in first order pde

somethingstra
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Hi guys,

I have a general problem that I'm not quite sure how to solve. Suppose you have a first order pde, like Ut=Ux together with some boundary conditions.

You'd do the appropriate transformations that lead to a solution plus an arbitrary function defined implicitly. How would you know that the solution is unique? Is there anyway to work with the initial conditions to figure that out?
 
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You know it is unique if the conditions for the "existance and uniqueness" theorem hold. What are those conditions?
 
Ah ok. Thanks
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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