It is important to know how to identify linear first order differential equations, because if you don't know whether an equation is linear first order differential equation or not then you will try to solve it in the wrong way.

I am not going to make a boring definition here. I will make a few typical examples so it will better help you understand the concept.

(1) Is the equation dy/dx = 6y⁴+9 a linear first order differential equation?

Answer: No. This is not because it doesn't involve only the first order terms in y and y', instead it has y⁴ in it.

(2) Is the equation dy/dx = 11y +2 a linear first order differential equation?

Answer: This is a linear first order differential equation, because it involves only the first oder terms in y and y'.

(3) Is this equation: dy/dx = x sin (y), a linear first order differential equation?

Answer: No. This is not, because it doesn't involve solely the first order terms in y and y'.