# Question on Ordinary Differential Equation (ODE)

## 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
Homework Helper
Gold Member
2020 Award
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
Gold Member
If you differentiate (1) wrt x, then
d/dx (du/dy) = d/dy (du/dy) dy/dx = d2u/dy2 dy/dx, by chain rule.