Difference between linear and non-linear first order DE

[oceanwalk]

Some really basic questions about first order DE, which I can't seem to get.

What is the difference between a linear and non-linear first order differential equation?

For example:
dy/dx = cos y

dy/dx = cos x

And if it's a non-linear first order DE, how do you tell whether it's seperable?

 

[lippyka]

A linear first order differential equation is one in which the highest derivative present is of the first order and the coefficients of the derivatives are constants. In contrast, a non-linear first order differential equation is one in which the highest derivative present is of the first order and the coefficients of the derivatives are functions of the independent variable and/or the dependent variable. In the given examples, the first equation dy/dx = cos y is a non-linear first order differential equation, while the second equation dy/dx = cos x is a linear first order differential equation. To tell whether a non-linear first order differential equation is separable, you need to check if it can be written in the form f(x)dx + g(y)dy = 0. If it can, then the equation is separable.