Editing Separation of variables (section)

== Ordinary differential equations (ODE) ==

A differential equation for the unknown <math>f(x)</math> is separable if it can be written in the form

:<math>\frac{d}{dx} f(x) = g(x)h(f(x))</math>

where <math>g</math> and <math>h</math> are given functions. This is perhaps more transparent when written using <math>y = f(x)</math> as:

:<math>\frac{dy}{dx}=g(x)h(y).</math>

So now as long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain:

:<math>{dy \over h(y)} = g(x) \, dx,</math>

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 [[Leibniz's notation]] may prefer to write this as

:<math>\frac{1}{h(y)} \frac{dy}{dx} = g(x),</math>

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 <math>x</math>, we have

{{NumBlk|:|<math>\int \frac{1}{h(y)} \frac{dy}{dx} \, dx = \int g(x) \, dx,  </math>|{{EquationRef|A1}}}}

or equivalently,

:<math>\int \frac{1}{h(y)} \, dy = \int g(x) \, dx </math>

because of the [[integration by substitution|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]] <math>\frac{dy}{dx}</math> 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 [[arbitrary constant of integration|constants of integration]], in equation ({{EquationNote|A1}}) as in
:<math>\int \frac{1}{h(y)} \, dy + C_1 = \int g(x) \, dx + C_2,</math>
because a single constant <math>C = C_2 - C_1</math> is equivalent.)

=== Example ===
Population growth is often modeled by the "logistic" differential equation

: <math>\frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)</math>

where <math>P</math> is the population with respect to time <math>t</math>, <math>k</math> is the rate of growth, and <math>K</math> is the [[carrying capacity]] of the environment.
Separation of variables now leads to

: <math>
\begin{align}
& \int\frac{dP}{P\left(1-P/K \right)}=\int k\,dt
\end{align}
</math>
which is readily integrated using partial fractions on the left side yielding

: <math>P(t)=\frac{K}{1+Ae^{-kt}}</math>

where A is the constant of integration. We can find  <math>A</math> in terms of <math>P\left(0\right)=P_0</math> at t=0. Noting <math>e^0=1</math> we get

: <math>A=\frac{K-P_0}{P_0}.</math>

=== Generalization of separable ODEs to the nth order ===
Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order or ''n''th-order ODE. Consider the separable first-order ODE:
:<math>\frac{dy}{dx}=f(y)g(x)</math>
The derivative can alternatively be written the following way to underscore that it is an operator working on the unknown function, ''y'':
:<math>\frac{dy}{dx}=\frac{d}{dx}(y)</math>
Thus, when one separates variables for first-order equations, one in fact moves the ''dx'' denominator of the operator to the side with the ''x'' variable, and the ''d''(''y'') is left on the side with the ''y'' variable. The second-derivative operator, by analogy, breaks down as follows:
:<math>\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}\left(\frac{d}{dx}(y)\right)</math>
The third-, fourth- and ''n''th-derivative operators break down in the same way. Thus, much like a first-order separable ODE is reducible to the form
:<math>\frac{dy}{dx}=f(y)g(x)</math>
a separable second-order ODE is reducible to the form
:<math>\frac{d^2y}{dx^2}=f\left(y'\right)g(x)</math>
and an nth-order separable ODE is reducible to
:<math>\frac{d^ny}{dx^n}=f\!\left(y^{(n-1)}\right)g(x)</math>

=== Example ===
Consider the simple nonlinear second-order differential equation:<math display="block">y''=(y')^2.</math>This equation is an equation only of ''y<nowiki>''</nowiki>'' and ''y'<nowiki/>'', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect all ''x'' variables on one side and all ''y'<nowiki/>'' variables on the other to get:<math display="block">\frac{d(y')}{(y')^2}=dx.</math>Now, integrate the right side with respect to ''x'' and the left with respect to ''y''':<math display="block">\int \frac{d(y')}{(y')^2}=\int dx.</math>This gives<math display="block">-\frac{1}{y'}=x+C_1,</math>which simplifies to:<math display="block">y'=-\frac{1}{x+C_1}~.</math>This is now a simple integral problem that gives the final answer:<math display="block">y=C_2-\ln|x+C_1|.</math>