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## Homework Statement

i)write down the general form of a semi lenear first order pde in the unknown u(x,y)

ii)write down the ode satisfied by a characteristic curve in the x-y plane for your pde

ii)give a careful derivation of the ode satisfied by u(x,y) along such a charcteristic curve.

## Homework Equations

## The Attempt at a Solution

i)[itex] a(x,y) \frac{\partial u}{\partial x} + b(x,y) \frac{\partial u}{\partial y} = g(x,y,u) [/itex]

ii) the characteristic traces are given by [itex]\frac{dx}{dt}[/itex] = a(x,y) and [itex]\frac{dy}{dt}[/itex] = b(x,y) and [itex]\frac{du}{dt}[/itex] = g(x,y,u) so is one of these what i'm looking for?

iii) since [itex]\frac{dx}{dt}[/itex] = a(x,y) along our characteristic we get t in term of x, provided a(x,y) [itex]\neq[/itex] 0. We can also express y in terms of x.

the chainrule gives[itex]\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}[/itex] and so [itex] \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{b(x,y)}{a(x,y)} [/itex] along the projected trace

=> the projected trace satisfies the ODE [itex]\frac{dy}{dx} = \frac{b(x,y)}{a(x,y)}[/itex]