All first order linear differential equations are solvable thanks to the existence of the integrating factor
But is it correct to say that all such DEs are separable? I'm not sure if the method involving the integral factors counts as solving by separation.
I ask because the "separable" adjective seems to be important in classifying DEs, and saying that all first order linear differential equations are separable is a pretty meaningful statement
Question: "all first order linear differential equations are separable is a pretty meaningful statement"
Answer: No, all first order linear differential equations are not separable.
A first order differential equation $~y’ = f\left( {x,y} \right)~$ is called a separable equation if the function $~f(x,y)~$ can be factored into the product of two functions of $~x~$ and $~y~$: $$f\left( {x,y} \right) = p\left( x \right)h\left( y \right),$$ where $~p(x)~$ and $~h(y)~$ are continuous functions.
There are differential equations those are linear and first order but not separable (counter example shown below).
...............................................................................
Question: "is it correct to say that all such DEs are separable ?"
Answer: It is not true that all first order linear differential equations those are solved by using the method of integrating factor are separable.
Example: Consider the differential equation $~\frac{dy}{dx}-\frac yx=4~$.
Clearly, the above differential equation is first order, linear but it cannot be factored into a function of just $~x~$ times a function of just $~y~$. So it is not separable.
It can be solve by using the method of integrating factor.
Solution: $$\frac{dy}{dx}-\frac yx=4\tag1$$Integrating factor (I.F.) is $~e^{-\int \frac 1x~dx}=\frac 1x~$.
Multiplying both side of equation $(1)$ by I.F. we have $$\frac 1x\cdot\frac{dy}{dx}-\frac y{x^2}=\frac 4x$$ $$\implies \dfrac{d}{dx}\left(\frac y{x}\right)=\frac 4x$$Integrating,$$\frac y{x}=4\ln x+c\implies y=4x\ln x~+~cx$$where $~c~$ is an integrating constant.
