# Thread: Implicit Differentiation

1. ## Implicit Differentiation

I am looking at a differential equations book and do not understand the mathematics behind the example:

1. Where did the derivative in (4) came from? I understand if we were to multiply both sides by eax.
2. How did was the math carried afterwards?

Thanks in advance.

Best

2. ## Re: Implicit Differentiation

Consider an ODE of the form:

$\displaystyle \frac{dy}{dx}+P(x)y=Q(x)$

Now, if we define an "integrating factor" as follows:

$\displaystyle \mu(x)=\exp\left(\int P(x)\,dx\right)$

Then it follows that:

$\displaystyle \frac{d\mu}{dx}=P(x)\mu(x)$

And so, multiplying through the ODE by this factor, there results:

$\displaystyle \mu(x)\frac{dy}{dx}+P(x)\mu(x)y=\mu(x)Q(x)$

And we may then write:

$\displaystyle \mu(x)\frac{dy}{dx}+\frac{d\mu}{dx}y=\mu(x)Q(x)$

At this point we should observe that:

$\displaystyle \frac{d}{dx}\left(\mu(x)\cdot y\right)=\mu(x)\frac{dy}{dx}+\frac{d\mu}{dx}y$

And so the ODE may be written:

$\displaystyle \frac{d}{dx}\left(\mu(x)\cdot y\right)=\mu(x)Q(x)$