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.
Ordinary differential equations (ODE) A differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form
d d x f ( x ) = g ( x ) h ( f ( x ) ) {\displaystyle {\frac {d}{dx}}f(x)=g(x)h(f(x))}
where g {\displaystyle g} and h {\displaystyle h} are given functions. This is perhaps more transparent when written using y = f ( x ) {\displaystyle y=f(x)} as:
d y d x = g ( x ) h ( y ) . {\displaystyle {\frac {dy}{dx}}=g(x)h(y).}
So now as long as h(y) ≠ 0, we can rearrange terms to obtain:
d y h ( y ) = g ( x ) d x , {\displaystyle {dy \over h(y)}=g(x)\,dx,}
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.
Alternative notation Those who dislike differentials as separate entities may prefer to write this as
1 h ( y ) d y d x = g ( x ) , {\displaystyle {\frac {1}{h(y)}}{\frac {dy}{dx}}=g(x),}
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 x {\displaystyle x} , we have
or equivalently,
∫ 1 h ( y ) d y = ∫ g ( x ) d x {\displaystyle \int {\frac {1}{h(y)}}\,dy=\int g(x)\,dx}
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 d y d x {\displaystyle {\frac {dy}{dx}}} 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 (A1) as in
∫ 1 h ( y ) d y + C 1 = ∫ g ( x ) d x + C 2 , {\displaystyle \int {\frac {1}{h(y)}}\,dy+C_{1}=\int g(x)\,dx+C_{2},}
because a single constant C = C 2 − C 1 {\displaystyle C=C_{2}-C_{1}} is equivalent.)
Example Population growth is often modeled by the "logistic" differential equation
d P d t = k P ( 1 − P K ) {\displaystyle {\frac {dP}{dt}}=kP\left(1-{\frac {P}{K}}\right)}
where P {\displaystyle P} is the population with respect to time t {\displaystyle t} , k {\displaystyle k} is the rate of growth, and K {\displaystyle K} is the carrying capacity of the environment. Separation of variables now leads to
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