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God damn constant of integration and DiffEQ

Casually HardcoreCasually Hardcore Once an Asshole. Trying to be better.Registered User regular
edited August 2009 in Help / Advice Forum
Okay, here's my question regarding first-order Differential Equations and separable equations.

Why is e^c = C ?

Is it because C is some constant? But wouldn't it matter that e is being raised by that constant?

Or is there some terminology that i'm missing?

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    ecco the dolphinecco the dolphin Registered User regular
    edited August 2009
    "C" has, in this case, been defined as just a shorthand way of writing "e^c".

    Basically, why write e^c when you can just write "C" in its place?

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    Casually HardcoreCasually Hardcore Once an Asshole. Trying to be better. Registered User regular
    edited August 2009
    Man, I'm thinking it's that simple. But there's no previous mentioning of this and this is in the solution manual. They're dropping Cs like nobodies business; superscripted, subscript, capital, small, etc. Maybe I should just read all these Cs as 'arbitrary constant' and ignore the formatting.

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    OrestesOrestes Registered User regular
    edited August 2009
    C represents constant in Calculus.

    So yeah...

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    shadydentistshadydentist Registered User regular
    edited August 2009
    ethugs4life wrote: »
    Man, I'm thinking it's that simple. But there's no previous mentioning of this and this is in the solution manual. They're dropping Cs like nobodies business; superscripted, subscript, capital, small, etc. Maybe I should just read all these Cs as 'arbitrary constant' and ignore the formatting.

    You've got it right. When you take an indefinite integral, you're left with an arbitrary constant, which you've called C. However, you could have just as easily called it ln(C), which would leave you with C if you take the exponent of it.

    Does that make sense?

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    ProPatriaMoriProPatriaMori Registered User regular
    edited August 2009
    Yep, e to any constant is another constant. Any constant is C. So C+C=C. C*C=C. C^C = C.

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    ronyaronya Arrrrrf. the ivory tower's basementRegistered User regular
    edited August 2009
    ... however, it's a bad habit to write e^C = C, because well it's wrong as you've pointed out.

    Just write =constant or just remove e^C implicitly when you need to.

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    Smug DucklingSmug Duckling Registered User regular
    edited August 2009
    It's because there's a small c and a big C. c not equal to C.

    e^c = C is just saying that c = ln( C) because it's easier to work with for some reason.

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    lunchbox12682lunchbox12682 MinnesotaRegistered User regular
    edited August 2009
    Are you sure the book isn't using a symbol that looks like "=" for the equation.
    It was a couple years ago, but thought mine used a "=" with a dot over it or some other pseudo equality symbol.

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    ronyaronya Arrrrrf. the ivory tower's basementRegistered User regular
    edited August 2009
    It probably isn't; writing e^c = C is a common way to reduce the visual complexity of an expression.

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    Teslan26Teslan26 Registered User regular
    edited August 2009
    Any combination of constants is a constant, and seeing as the c is just as unknown as the C - may as well combine it into one arbitrary constant instead of having 2 or more to deal with.

    i.e. k(1).e^c + k(2) = C

    When you solve you either need to find c and k(1) and k(2), or just C. C is way simpler and faster.

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