A differential equation if "linear" if it does not involve any non-linear functions of the dependent variable. It is linear if it can be written as

a_0(t)y^n+a_1(t)y^n^-^1+...+a_n(t)y=g(t) then it is linear where y(t) is the unknown.

I have been trying to understand the difference between linear and nonlinear differential equations. If one has some random differential equation of the form

then, if I am correct, it is first order because it is the first derivative and ordinary because there is only one independent variable, being x, correct?

Now suppose the answer is known to be some rational expression of the form

where P(x) and Q(x) are polynomials, and let's say they are of the first degree, meaning the highest order of x is to the first power.

Is the original differential equation linear or nonlinear? The result is not a linear function and has nothing to do with transcendentals so far as I am aware. I would think that the original equation would be called nonlinear but I wanted to make sure.

Thanks.

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