Linear basically means it has the same properties as a line. The easiest classification of this is that the equation only contain terms that are singular powers (i.e., ).

So is not linear

is also not linear.

however, is linear. (no powers, exponentials, logarithms, roots, just plain old ordinary variables)

In general, if we haves variables , then a linear equation is of the form.

Where and d are all constants.

Now a first order ordinary differential equation is always of this form

where f(y) is some expression involving y, as a function of x.

So, is a differential equation.

and is also a differential equation.

We call a differential equation linear, if f(y) is linear. So the first one is linear, but the second one isn't.

How the book got it's answers was through simple algebra. In order to determine if the differential equation is linear in x, we needed the derivative of x on one side and an expression involving x on the other.

So, we do this

Since the right hand side is a linear expression, we say the differential equation is linear in x.

However, similar algebra tells us that the differential equation is NOT linear in y.

i.e., . Since the right hand side is not linear, the differential equation is not linear.