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In my introductory differential class, the professor stated that there exist some ODEs that are both linear and separable.

I'm a bit confused, because I was under the impression that a separable differential equation is a type of nonlinear differential equation.

Can an ODE be both separable and linear? If so please provide a basic example. I could not find anything that answered my question on google.

Edit: Many of you have provided me with examples of functions that are both linear and separable. I read on Lamar.edu "We are now going to start looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations that we will look at is separable differential equations." I thought that meant that any of the examples we provided are trivially separable, and should be treated as linear only.

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    $\begingroup$ How about $x'=x$. $\endgroup$ – Alex S Sep 24 '15 at 16:05
  • $\begingroup$ Thank you for your example. I thought that examples like that might be trivial, but I guessed wrong. $\endgroup$ – JohnKraz Sep 24 '15 at 17:26
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In fact, linear ODEs that are immediately separable have a name: linear homogeneous equations. Their form is: $$y'=f(x) \, y$$ that can be directly integrated from $$\frac{dy}{y}=f(x)\, dx$$

In contrast, nonhomogeneous linear ODEs $$y'=f(x) \, y + g(x)$$ cannot be separated in general, although there exists an algorithmic procedure to solve them by quadrature.

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Yes, like $$ y'(x)+g(x)y(x)=0. $$

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  • $\begingroup$ Thank you for your example. $\endgroup$ – JohnKraz Sep 24 '15 at 17:25
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$$y' = 1$$ is certainly linear and can be rewritten as $$dy = dx$$ so it is separable as well...

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  • $\begingroup$ Thank you for your example. $\endgroup$ – JohnKraz Sep 24 '15 at 17:26

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