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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 nonseparable differential equations with the help of solved examples for a better understanding.
What are Separable Differential Equations?
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 abovegiven forms of separable differential equations are equivalent. A differential equation in the above form can be solved using the method separation of variables.
Separable Differential Equations Definition
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 = (x^{2} + 6)(y  7)
 y' = cos x sec y
 dy/dx = ye^{x}
 y' = xy  3x + 4y  12
 dy/dx = sin y
Solving Separable Differential Equations
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 = x^{2}/2  x + C_{1}, where C_{1} 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
Initial Value Problem Separable Differential Equations
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)(y^{2}  9), y(0) = 1
Solution: dy/dx = (x  2)(y^{2}  9)
⇒ dy/(y^{2}  9) = (x  2)dx
⇒ ∫ (1/(y^{2}  9)) dy = ∫(x  2)dx
⇒ (1/6) ∫ [1/(y  3)]  [1/(y + 3)] dy = x^{2}/2  2x + C_{1} [Using integration method of partial fractions]
⇒ (1/6) [ln y  3  ln y + 3] = x^{2}/2  2x + C_{1}
⇒ ln y  3  ln y + 3 = 3x^{2}  12x + C_{2}, [6C_{1} = C_{2}]
⇒ ln(y  3)/(y + 3) = 3x^{2}  12x + C_{2}
⇒ (y  3)/(y + 3) = e^{3x2  12x + C2}
⇒ (y  3)/(y + 3) = C_{3} e^{3x2  12x} [C_{3 }= ± e^{C2} as we have removed the absolute sign]  (1)
⇒ y  3 = C_{3 }(y + 3) (e^{3x2  12x})
⇒ y  3 = y (C_{3}e^{3x2  12x}) + 3 (C_{3 }e^{3x2  12x})
⇒ y (1  C_{3 }e^{3x2  12x}) = 3 (1 + C_{3 }e^{3x2  12x})
⇒ y = 3 (1 + C_{3 }e^{3x2  12x})/(1  C_{3 }e^{3x2  12x})
Now, to determine the value of C_{3}, 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) = C_{3} e^{3x2  12x}
⇒ (1  3)/(1 + 3) = C_{3} e^{3(0)2  12(0)}
⇒ 2 = C_{3}
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
Separable Differential Equations Examples

Example 1: Solve the initial value problem of separable differential equations: dr/dθ = r^{2}/θ, r(1) = 2.
Solution: The given differential equation is a separable differential equation. Hence, we have
dr/dθ = r^{2}/θ
⇒ dr/r^{2} = dθ/θ
⇒ ∫ dr/r^{2} = ∫ dθ/θ
⇒ 1/r = ln θ + C
Applying the initial condition r(1) = 2, we have
1/2 = ln 1 + C
⇒ 1/2 = 0 + C
⇒ C = 1/2
Therefore, we have 1/r = ln θ  1/2
⇒ r = 1/(1/2  ln θ)
Answer: The required solution is r = 1/(1/2  ln θ)

Example 2: Check if the differential equation y' = xy  21 + 3y  7x is separable.
Solution: We have
y' = xy  21 + 3y  7x
⇒ dy/dx = xy  7x + 3y  21
⇒ dy/dx = x(y  7) + 3(y  7)
⇒ dy/dx = (x + 3) (y  7)
Since the given differential equation can be written as dy/dx = f(x) g(y), where f(x) = x + 3 and g(y) = y 7, therefore it is a separable differential equation.
Answer: y' = xy  21 + 3y  7x is a separable differential equation.
FAQs on Separable Differential Equations
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 NonSeparable Differential Equations?
Differential equations that cannot be written as dy/dx = f(x) g(y) are nonseparable differential equations. In other words, we can say those differential equations which are not separable are called nonseparable differential equations.
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