What is Separable Differential Equation?

All the differential equations in which we can separate the variable from each other are called the separable differential equation the general form of the separable differential equation is dy/dx = f(x) g(y).

We can easily solve these types of equations by simply separating the variables in the differential equation and then integrating them individually.

All the ways in which we can write the separable differential equation are,

• dy/dx = f(x) g(y)
• dy/ g(y) = dx f(x)
• dy/dx = f(x)/g(y)
• g(y)dy = f(x) dx

Variable Separable Differential Equation Definition

Variable separable differential equation is defined as the equation of the form dy/dx = f(x) g(y), where f(x) and g(x) are the functions of the x and y. The solution to these equations is achieved by separating the variables and then integrating them separately.

Examples of Separable Differential Equation

Various examples of separable differential equations are,

• dy/dx = (2x³ + 6)(y² – 7)
• dy/dx = sec x cosec y
• dy/dx = e^ye^x
• dy/dx = sin y cos x

Identify Separable Equation

The first-order differential equation is a separable differential equation if and only if it can be written as

\frac{dy}{dx} = f(x).g(y)

where,

• f(x) is a function of x that does not contain y
• g(y) is a function of y that does not contain x

If it is not possible, to write the equation in this form we call the equation not a separable differential equation.

Note: In order to solve this type of differential equation we have to separate all y’s on one side and x’s on another side of the equal sign.

Separable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y’ = f(x).g(y), is called a separable differential equation.

Basic form of the Separable differential equations is dy/dx = f(x) g(y), where x is the independent variable and y is the dependent variable.