Why is the characteristic of (d/dx) + (d/dt) = 0 not c = x + t?

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    2nd order Pde
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The discussion clarifies that the characteristic equation (d/dx) + (d/dt) = 0 leads to the conclusion that the characteristic lines are defined by c = x - t, rather than c = x + t. This is established through the analysis of the function forms, where the equation (\partial_{x}+\partial_{t})f(x+t) results in 2f'(x+t), while (\partial_{x}+\partial_{t})f(x-t) simplifies to zero. Therefore, the solutions must take the form u(x,t) = f(x-t), indicating that x - t = constant represents the correct characteristic lines.

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Why is the characteristic of

(d/dx) + (d/dt) = 0 where d is small delta

c = x - t and not c = x + t
 
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Because(\partial_{x}+\partial_{t})f(x+t)=2f'(x+t)

While

(\partial_{x}+\partial_{t})f(x-t)=f'(x-t)-f'(x-t)=0

So you need solutions of the latter form u(x,t)=f(x-t), which means x-t=const are charcteristic lines.
 

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