Variable Separable Method

If an equation is of form

dydx+xy=0

And it can be written in such a way where the terms of x are written along with dx and the terms of y are written along with dy, i.e.,

y.dy+x.dx=0

Then the equation can be termed as a variable separable equation and it can be solved through a variable separable method. For example, if the equation has two variables x and y, the equation can be written such that all the x terms are on one side along with dx and all the y terms are along with dy.

Suppose we are given an equation  dydx=(x+y)2. In this equation, no matter how hard we try or what method we use, we can’t separate the variables for getting the terms of x with dx and that of y with dy. Hence, these types of equations aren’t variable separable.

Variable separable method is used for both ordinary and partial differential equations. For ordinary differential equations, first, the variables are explicitly separated and written with their respective derivative, then we simply integrate the equation formed to get the required solution. For partial differential equations, the variables are separated with the help of some simple substitutions making the equation a combined form of different ordinary differential equations and then integrating them to get the required result.

Separable differential equation:

If any equation of general form 

dydx=f(x,y)

Can be written as a separation of variables, i.e. the variables of the equation, x and y, can be explicitly separated from one another, then it’s a separable differential equation. The variable separable form of the above used general form of differential equation is

fx,y=gx.h(y)

Here, g and h are continuous functions of a particular interval.

How to apply a variable separable method?

It has been discussed above that which equations are separable differential equations. Now, we are going to learn how to solve the variable separable equations using the variable separable method.

Considering the example we used above in the introduction,

dydx+xy=0

In this equation, the variables aren’t separated yet. So, making it variable separable, it can be written as

y.dy+x.dx=0

y.dy=-x.dx

Now, integrating both sides of the equation with respect to x,

y.dy=-x.dx

y.dy=-x.dx

y22=-x22+c

It is the required solution of the differential equation, where c is an arbitrary constant.

Let’s see some examples on the application of variable separable method-

• dydx=e3x-y+x2e-y

Solution: Let’s rewrite the equation in variable separable form,

dydx=e-y(e3x+x2)

dye-y=e3x+x2dx

eydy=e3x+x2dx

Integrating both sides of the equation, we get

eydy=e3x+x2dx

ey=e3x3+x33+c

This is the required solution of the given differential equation.

• 3y.dy=x4+1dx

Solution: Here, the given equation is already in a variable separable form.

Integrating both sides of the equation, we get

3y.dy=(x4+1)dx

y3=x55+x+c

This is the required solution of the given differential equation.

• tan x .y.dx+x.cot y.dy=0  

Solution: The equation can be rewritten as

tan x .y.dx+x.cot y.dy=0  

Integrating both sides of the equation,

x 2-y 2=c

This is the required solution of the equation.

Initial value problem Separable differential equations:

We just discussed the general solution of variable separable differential equations. Now, we are going to discuss the separable differential equations of the initial value problem. In these problems there is a variable separable equation given with some initial fixed values of variables of the equation, which we have to use while solving the equation. An example of an initial value problem is  dydx=x-3y2-8, y0=-1.

Applications of variable separable equation:

Variable separable equations help in many physical situations such as- 

Newton’s law of cooling: It states that the rate at which the object’s temperature is changing is proportional to the difference between the temperature of its surroundings and its own temperature.

Solution concentrations: We can use the variable separable method to determine the concentration of a particular substance in a solution with respect to time.

Conclusion:

The variable separable method is used to solve for a general solution of separable differential equations, which are the equations whose variables can be explicitly separated. There are some equations that are not directly variable separable; for such equations, we have to make a simple substitution to solve them, such as homogeneous differential equations. Before starting this, you should know the basics of integration, as solving variable separable equations involves lots of integration.