I'm having trouble understanding this for some reason...am I right about these?
xy' - 4y = 0 is linear.
y' = y - x^3 is linear.
x' + 2x - e^(-x) = 0 is non-linear.
An ODE is linear if it is of the form $\displaystyle a(x)y^{(n)} + b(x)y^{(n - 1)} + ~...~ + c(x)y = F(x)$. Where $\displaystyle y^{(n)}$ is the nth derivative of y(x) and all the a(x), b(x), ...., c(x), F(x) are all functions of x only.
Note that in the first two you are thinking of "x" as the independent variable- y is a function of x. But in the last, x is the dependent variable- x is a function of some unnamed variable.