Editing Separation of variables (section)

=== 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.)