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Math Help - Partial Differential Question

  1. #1
    Junior Member
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    Partial Differential Question

    Hi! Haven't been on here for ages.

    I'm a little confused about part of a question I'm doing. I won't write it all out, but basically u=u(x(s,r), t(s,r))

    \displaystyle \frac{\partial u}{\partial s} = -u

    \displaystyle \frac{\partial x}{\partial s} = 3x

    \displaystyle \frac{\partial t}{\partial s} = 1

    From that we supposedly get t = s + A(r) which I do understand, but also:
    x=B(r)exp(3s) and u=A(r)exp(-s).

    Maybe I'm a bit too tired to be doing this, but where do the exponentials come from?

    Thanks!
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    They are just solving the other equations

    \displaystyle \frac{\partial x}{\partial s}=3x \iff \frac{dx}{x}=3ds \iff \ln|x|=3s+c \iff x(s,r)=Ce^{3s}

    The C is a constant of integration, but it is only constant with respect to s not r because we used partial integration to undo the partial derivative.

    The u equation is solved exactly the same way! your initial conditions will define the arbitrary constants(functions)
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