Separable Differential Equations

Separable differential equations are a special type of differential equations where the variables involved can be separated to find the solution of the equation. Separable differential equations can be written in the form dy/dx = f(x) g(y), where x and y are the variables and are explicitly separated from each other. After separating the variables, the solution of the differential equation can be determined easily by integrating both sides of the equation. The separable differential equation dy/dx = f(x) g(y) is written as dy/g(y) = f(x) dx after the separation of variables.

In this article, we will understand how to solve separable differential equations, initial value problems of the separable differential equations, and non-separable differential equations with the help of solved examples for a better understanding.

Differential equations in which the variables can be separated from each other are called separable differential equations. A general form to write separable differential equations is dy/dx = f(x) g(y), where the variables x and y can be separated from each other. Some other forms of separable differential equations are given below which will help to identify them while solving problems:

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

Please note that all the above-given forms of separable differential equations are equivalent. A differential equation in the above form can be solved using the method separation of variables.

A separable differential equation is defined to be a differential equation that can be written in the form dy/dx = f(x) g(y). This implies f(x) and g(y) can be explicitly written as functions of the variables x and y. As the name suggests, in the separable differential equations, the derivative can be written as a product the function of x and the function of y separately. We can check if a differential equation is separable by checking if the derivative dy/dx can be expressed as a function of x times the function of y.

Some of the examples of separable differential equations are given below:

• dy/dx = (x² + 6)(y - 7)
• y' = cos x sec y
• dy/dx = ye^x
• y' = xy - 3x + 4y - 12
• dy/dx = sin y

Now that we know how to identify separable differential equations, we will learn how to solve them. We will solve a separable differential equation to understand the process of solving them. To solve such differential equations, follow the basic steps given below:

• Step 1: Write the derivative as a product of functions of individual variables, i.e., dy/dx = f(x) g(y)
• Step 2: Separate the variables by writing them on each side of the equality, i.e., dy/g(y) = f(x) dx
• Step 3: Integrate both sides and find the value of y, and hence the general solution of the separable differential equation, i.e., ∫ dy/g(y) = ∫ f(x) dx

Example: Consider a separable differential equation dy/dx = xy + 2 - 2x - y

First, we will write xy + 2 - 2x - y as a product of the function of x and the function of y.

dy/dx = xy + 2 - 2x - y

⇒ dy/dx = (x - 1)(y - 2)

⇒ dy/(y - 2) = (x - 1) dx [Separating the variables]

⇒ ∫dy/(y - 2) = ∫(x - 1) dx [Integrating both sides]

⇒ ln |y - 2| = x²/2 - x + C₁, where C₁ is constant of integration

⇒ y - 2 = e^{x2/2 - x + C₁}

⇒ y = 2 + Ce^{x2/2 - x}, where e^C₁ = C

Hence, y = 2 + Ce^{x2/2 - x} is the general solution of the separable differential equation dy/dx = xy + 2 - 2x - y

We have learned to find the general solution of separable differential equations. Next, we will solve initial value problems involving separable differential equations which are given as dy/dx = f(x) g(y), y(x_o) = y_o, where y_o is a fixed value of y at x = x_o. Let us solve an example to understand its application and find a particular solution.

Example: Solve the separable differential equation dy/dx = (x - 2)(y² - 9), y(0) = -1

Solution: dy/dx = (x - 2)(y² - 9)

⇒ dy/(y² - 9) = (x - 2)dx

⇒ ∫ (1/(y² - 9)) dy = ∫(x - 2)dx

⇒ (1/6) ∫ [1/(y - 3)] - [1/(y + 3)] dy = x²/2 - 2x + C₁ [Using integration method of partial fractions]

⇒ (1/6) [ln |y - 3| - ln |y + 3|] = x²/2 - 2x + C₁

⇒ ln |y - 3| - ln |y + 3| = 3x² - 12x + C₂, [6C₁ = C₂]

⇒ ln|(y - 3)/(y + 3)| = 3x² - 12x + C₂

⇒ |(y - 3)/(y + 3)| = e^{3x2 - 12x + C₂}

⇒ (y - 3)/(y + 3) = C₃ e^{3x2 - 12x} [C₃ = ± e^C₂ as we have removed the absolute sign] --- (1)

⇒ y - 3 = C₃ (y + 3) (e^{3x2 - 12x})

⇒ y - 3 = y (C₃e^{3x2 - 12x}) + 3 (C₃ e^{3x2 - 12x})

⇒ y (1 - C₃ e^{3x2 - 12x}) = 3 (1 + C₃ e^{3x2 - 12x})

⇒ y = 3 (1 + C₃ e^{3x2 - 12x})/(1 - C₃ e^{3x2 - 12x})

Now, to determine the value of C₃, we will put the initial value into the general solution of the separable differential equation. We can put the initial value into another equivalent equation of the general equation which is equation (1). Therefore, we have

(y - 3)/(y + 3) = C₃ e^{3x2 - 12x}

⇒ (-1 - 3)/(-1 + 3) = C₃ e^{3(0)2 - 12(0)}

⇒ -2 = C₃

Therefore, the solution of the initial value problem is y = 3(1 - 2 e^{3x2 - 12x})/(1 + 2 e^{3x2 - 12x})

Important Notes on Separable Differential Equations

• Some of the common applications of separable differential equations are Newton's Law of Cooling, Determining solution concentration, etc.
• General form of separable differential equation is y' = f(x) g(y)
• The method that is used to solve separable differential equations is called the method of separation of variables.

Related Topics on Separable Differential Equations

• Rules of Differentiation
• Differentiation and Integration Formula
• Product Rule Formula
• Chain Rule Formula

What are Separable Differential Equations in Calculus?

Differential equations in which the variables can be separated from each other are called separable differential equations. A general form to write separable differential equations is dy/dx = f(x) g(y), where the variables x and y can be separated from each other.

How to Identify Separable Differential Equations?

Any differential equation which can be written in any of the following forms is a separable differential equation:

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

How to Solve Separable Differential Equations?

To solve separable differential equations, we can follow the basic steps given below:

• Step 1: Write the derivative as a product of functions of individual variables, i.e., dy/dx = f(x) g(y)
• Step 2: Separate the variables by writing them on each side of the equality, i.e., dy/g(y) = f(x) dx
• Step 3: Integrate both sides and find the value of y, and hence the general solution of the separable differential equation, i.e., ∫ dy/g(y) = ∫ f(x) dx

How Do you Know if a Differential Equation is Separable?

A differential equation that can be written as dy/dx = f(x) g(y) is a separable differential equation. Hence, we can check if the differential equation can be written in the given form.

What is the Difference Between Linear and Separable Differential Equations?

Linear differential equations are differential equations where the degree of the derivative dy/dx is 1 and no other derivatives of higher power appear in the differential equation. On the other hand, a separable differential equation is defined to be a differential equation that can be written in the form dy/dx = f(x) g(y).

What are Non-Separable Differential Equations?

Differential equations that cannot be written as dy/dx = f(x) g(y) are non-separable differential equations. In other words, we can say those differential equations which are not separable are called non-separable differential equations.