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

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

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

Separation of variables in ODE_compressed

**Separation of variables in ODE**

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

In case if it would have any term like

**Step 1**

Now I will separate

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

Now I will get the value of

And here

Now I substitute back

Hence I can rewrite equation (3) as

At the end, I will put the values of

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

So the conclusion is

Now I will go to the second example.

**Example 2**

According to Stroud and Booth (2013)* “Solve

**Solution**

Here the given differential equation is:

Like my first example, here also I will separate

**Step 1**

First of all, I will rewrite the equation as

Next, I’ll take out

Now I will divide the equation through out with

So this gives

And this is only because

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

Then I’ll simplify it to get

Also, I already know that

Then I’ll simplify it to get the value of

So this means

Next I substitute this value of

Then I’ll simplify it to get

But I can also rewrite it as

Hence I can conclude that

Now I’ll give another example.

**Example 3**

According to Stroud and Booth (2013)* “Solve

**Solution**

Here the given differential equation is:

Like the first examples, here also I will separate

**Step 1**

First of all, I will rewrite the equation as

Next, I’ll rewrite

So the differential equation becomes

Next, I’ll separate the

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

Thus I can say that

So this means I have to get the value of

Therefore I’ll substitute

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

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!!

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