Question on Ordinary Differential Equation (ODE)

[a150daysflood]

Homework Statement

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

Homework 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))

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?

 

[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.

 

[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.