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Concept
Separation of variables
Summary
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
A differential equation for the unknown will be separable if it can be written in the form
where and are given functions. This is perhaps more transparent when written using as:
So now as long as h(y) ≠ 0, we can rearrange terms to obtain:
where the two variables x and y have been separated. Note dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential (infinitesimal) is somewhat advanced.
Those who dislike Leibniz's notation may prefer to write this as
but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to , we have
or equivalently,
because of the substitution rule for integrals.
If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.
(Note that we do not need to use two constants of integration, in equation () as in
because a single constant is equivalent.)
Population growth is often modeled by the "logistic" differential equation
where is the population with respect to time , is the rate of growth, and is the carrying capacity of the environment.
Separation of variables now leads to
which is readily integrated using partial fractions on the left side yielding
where A is the constant of integration. We can find in terms of at t=0. Noting we get
Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order or nth-order ODE.
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