1. ## 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))

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

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

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

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!

2. They are just solving the other equations

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

The $\displaystyle C$ is a constant of integration, but it is only constant with respect to $\displaystyle s$ not $\displaystyle r$ 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)