Definition:Separable Differential Equation
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
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.7$: Homogeneous Equations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(2)$ Variables separable
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(2)$ Variables separable
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): separable first-order differential equation
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): separable first-order differential equation