Question

Is every homogenous linear differential equation separable?

Answer 1

That depends on the order, for instance:
\[\large y'''-5y''-22y'+56y=0 \]
is a linear 3rd order differential equation, it is not separable though, you solve it via substitution, solving the polynomial form and back substitution.

Answer 2

How about in the 1st order case only?

Answer 3

A first order differential equation is of the form
\[\large y'+a(x)y=g(x) \]
we set \(g(x)=0\) to make it homogenous, so you're left with:
\[\frac {dy}{dx}+a(x)y=0 \implies \frac{1}{y}dy=-a(x)dx, \ \forall y \neq 0 \]

Answer 4

So it is possible?

Answer 5

yes, the above equation is separated, however you're not granted that you can solve the integration.

Answer 6

Like you can continue with the above manipulation and integrate both sides to get a explicit formula for the solution. Integration of both sides leads you to: \[ \large \log |y| = - \int a(x)dx+C \implies |y| = e^C \cdot e^{-\int a(x)dx} \] since \(e^C > 0\) and we have said that \(y \neq 0 \) you can drop the absolute value bar and introduce a new constant, so you will be left with:
\[\large y= Ke^{- \int a(x) dx} \] but same as before, you're not granted to be able to solve this equation even though you can separate it. So the question might should be, is every homogenous, first order linear differential equation separable? Yes it is.

Is any first order linear differential equation solvable through separation, no it is not.