Solve the first-order ODE using transformations. Hello friends, today I’ll show how to solve the first-order ODE using transformations of variables. Have a look!!

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

**Solve First-order homogeneous equations**

**Separation of variables in the first-order ODE**

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**How to solve the first-order ODE using transformations of variables?**

Now there is a standard method to solve any first-order ODE using transformations of variables.

That is,

- First of all, investigate the equation.
- Next, look for an expression which is present more than once.
- After that, choose that expression as equals to any other variable, say, .
- Then replace that expression and with and respectively.
- Hence solve the equation.
- Finally, bring back the original expression.

In order to explain myself a bit better, I will solve an example on the first-order ODE using transformations of variables.

**Example on how to solve the first-order ODE using transformations of variables**

Disclaimer: None of these examples is 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 example.

**Example **

According to Stroud and Booth (2013)* “By substituting solve the equation of given that when ”

**Solution**

Here the given differential equation is:

I can rewrite the equation as

Now, this is not a homogeneous equation. This is because it has a constant term .

So, my first task is to transform the variables.

For that, I’ll transform . This is because is present in this equation more than once.

**Step 1**

Let me choose

Now I’ll differentiate with respect to . And that gives

Therefore I can rewrite in terms of as

So this gives

Next, I’ll replace with and with in the original equation.

Thus it will be

Now I’ll simplify it further.

So, first of all, I’ll cancel from both sides.

Thus it will be

Next, I’ll separate the two variables and .

Therefore it will look like

And that means

So this gives

Thus I can say that

Now I can easily separate the two variables and . Hence it will be

(1)

Then I’ll integrate both sides of the equation (1) to get the solution of the equation.

**Step 2**

Thus it will be

Now here is the integration constant. Next, I’ll replace with again.

So it will be

(2)

Then I’ll substitute in the equation (2) to get

After that I’ll simplify it to get

So this gives

which means

Now I put back in the equation (2) to get

Next, I’ll simplify it a bit more to get the particular solution of the given equation. Thus it will be

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

And this is the answer to this 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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