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Thread: Implicit Differentiation

  1. #1
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    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.

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  2. #2
    MHF Contributor MarkFL's Avatar
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    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:

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