Separation of variables in ODE. Dear friends, today I will show how to use the ‘separation of variables’ method in ordinary differential equations.

### Separation of variables in ODE

If you are looking for more in the first-order ODE, do check in:

**Solve first order ODE using ‘transformations of variables’**

As the name says, in this method, the variables are separated first.

Then both sides are integrated to get the solution to the equation.

Now I will give you three examples of ‘separation of variables’ method. Have a look!!

#### Examples of separation of variables

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2013)* “Find the general solution of ”

**Solution**

Here the given equation is

This equation does not have any term like . Here can be real number except zero.

This means in this example it will be possible to separate and variables.

In case if it would have any term like or , then the ‘separation of variables’ method would not work.

###### Step 1

Now I will try to separate and components of this equation.

So for this, I will divide the given differential equation with . Thus I get

Next, I’ll simplify it to get

Now I will integrate both sides of .

###### Step 2

Thus it will be

(1)

From the standard formula of integration, I already know that

Thus the left-hand side of the equation (1) will be

(2)

Now I will work on the right-hand side of the equation.

(3)

From the standard rule of differentiation, I already know that if ,

In other words, I can also say that

Thus I can rewrite as

Now I will get the value of . So it will be

And here is the integration constant.

Now I substitute back in to get

Hence I can rewrite equation (3) as

At the end, I will put the values of and equation (2) in equation (1). So it becomes

Now I’ll simplify it a bit. So this gives

So the conclusion is is the solution of the equation.

Now I will go to the second example.

##### Example 2

According to Stroud and Booth (2013)* “Solve given when ”

##### Solution

Here the given differential equation is:

Like my first example, here also I will separate and components.

###### Step 1

First of all, I will rewrite the equation as

Next, I’ll take out as a common factor like

Now I will divide the equation through out with to get

So this gives

And this is only because is not equal to . Now I’ll separate the and variables. So this gives

Next, I will integrate both sides of the equation.

###### Step 2

Thus the equation will look like

(4)

Now from the standard formula of integration, I already know that

Thus the left hand side of the equation (4) will be

(5)

Also I’ll use the standard formula of integration for the right hand side of the equation (4). Thus it will be

(6)

Next, I will put the values from (5) and (6) to (4) to get the solution of the given differential equation as

(7)

Well, this is the general solution of the equation.

Now I have to get the particular solution of the equation for .

###### Step 3

So I put and in the equation (7). And this gives

Then I’ll simplify it to get

Also, I already know that . Thus it gives

Then I’ll simplify it to get the value of as

So this means

Next I substitute this value of in the equation (7) to get the particular solution of the given equation. Thus it gives

Then I’ll simplify it to get

But I can also rewrite it as

Hence I can conclude that is the solution of the given equation for

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)* “Solve , given that when .”

**Solution**

Here the given differential equation is:

Like the first examples, here also I will separate and components.

###### Step 1

First of all, I will rewrite the equation as

Next, I’ll rewrite as . And this is because, as per the law of indices .

So the differential equation becomes

Next, I’ll separate the and component of the equation. So this gives

Now I’ll solve this equation. And for that, I’ll integrate this equation.

###### Step 2

Thus it gives

As per the standard laws of integration, .

Hence this equation becomes

And here is the integration constant.

Thus I can say that is the general solution of the equation. Now I have to get the particular solution of the equation for .

So this means I have to get the value of for .

Therefore I’ll substitute to the general solution of the given differential equation. So this gives

Next, I’ll simplify it to get the value of as

Thus the particular solution to the given differential equation is

As I can see, it is still possible to simplify it a bit. So this gives

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

## Leave a Reply