Question on Ordinary Differential Equation (ODE)

1. Aug 25, 2013

a150daysflood

1. The problem statement, all variables and given/known data

Find the ODE of the following
(1) du/dy = -u
(2) d^2u/dxdy = -du/dx

2. Relevant equations

For question 1, the answer is u= A(x)e^(-y)
while for question 2, the answer is u= e^(-y)(B(X) + c(Y))

3. The attempt at a solution

I've already solved the question, but just want to ask, if i change (1) to (2), its just a differentiation with respect to dx, but yet the solution contain a c(Y) term, any idea why is this so?

2. Aug 25, 2013

haruspex

To get those answers, these are PDEs, surely?
In (2), both sides have been differentiated partially wrt x. So any additive term that depends solely on y will have disappeared and cannot be reconstructed from the PDE. It's analogous to the constant of integration in ODEs.

3. Aug 25, 2013

CAF123

If you differentiate (1) wrt x, then
d/dx (du/dy) = d/dy (du/dy) dy/dx = d2u/dy2 dy/dx, by chain rule.