7 - Linear Differential Equations and The Integrating Factor

So, we have learned that we solve differential equations by integrating after moving around variables and everything to get a general form we can solve.

We will now learn about another very important type of differential equations called Linear Differential Equations. These are equations that we can't just separate.

Take a look at this:You might be thinking: "Oh boy."

Well, it's actually much easier than it seems.
In these types of scenarios, the first thing you will want to do is try as hard as you can to get the derivative of y alone.

This is the general form of the linear differential equation. Where P(x) is some kind of function involving x and Q(x) is the rest of the equation, dependent on the variable x.

This P(x) is very important.

Here's an idea. Look at this in another form: Recall a familiar pattern:
This is the product rule.

Yeah! Our strategy is to first make such equation solvable by completing this product rule.
We do this by using the integrating factor.
We need a function that yields itself, and that repeats itself all over when differentiated or integrated. And, oh yeah, that's the one and only:

I'm not going to get into the details as to how or why, but, to get what you need, you will use this P(x) in the concept of the integrating factor.

We define a function: Try to calculate on your own.

Answer: We have our rho. What you will do next is multiply the entire equation after making sure the derivative is alone. We get:Now that's familiar. Try to differentiate . y and its derivative are already there in their simplest forms.

Our product rule is now complete.

You now simply need to integrate.The integral of the derivative is just the original function.
All you need to do now is move to the other side.

You have now solved your first Linear Differential Equation!
Great job.

8 - Recap on The Fundamentals of Differential Equations

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