Definition:Separable Differential Equation

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Definition

A first order ordinary differential equation which can be expressed in the form:

$\dfrac {\d y} {\d x} = \map g x \map h y$

is known as a separable differential equation.


Its general solution is found by solving the integration:

$\ds \int \frac {\d y} {\map h y} = \int \map g x \rd x + C$


General Form

A first order ordinary differential equation which can be expressed in the form:

$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$

is known as a separable differential equation.


Its general solution is found by solving the integration:

$\ds \int \frac {\map {g_1} x} {\map {g_2} x} \rd x + \int \frac {\map {h_2} y} {\map {h_1} y} \rd y = C$


Also presented as

Some sources present this as an equation in the form:

$\dfrac {\d y} {\d x} = \dfrac {\map g x} {\map h y}$

or:

$\map h y \dfrac {\d y} {\d x} = \map g x$

whose general solution is found by solving the integration:

$\ds \int \map h y \rd y = \int \map g x \rd x + C$


Other sources have:

$\map g x + \map h y \dfrac {\d y} {\d x} = 0$

whose general solution is found by solving the integration:

$\ds \int \map g x \rd x = -\int \map h y \rd y + C$


Also known as

A separable differential equation is also known as a differential equation with separable variables.

Some sources refer to it as a variables separable differential equation.


Examples

Arbitrary Example $1$

Consider the first order ODE:

$(1): \quad \map {\dfrac \d {\d x} } {\map f x} = 3 x$

where we are given that $\map f 1 = 2$.

The particular solution to $(1)$ is:

$\map f x = \dfrac {3 x^2 + 1} 2$


Arbitrary Example $2$

Solution to Separable Differential Equation/Examples/Arbitrary Example 2

Also see

  • Results about separable differential equations can be found here.


Historical Note

The method of solution of a separable differential equation was described by Johann Bernoulli between the years $\text {1694}$ – $\text {1697}$.


Sources

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