God damn constant of integration and DiffEQ

Casually Hardcore

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?

ecco the dolphin

"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?

Casually Hardcore

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.

Orestes

C represents constant in Calculus.

So yeah...

shadydentist

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?

ProPatriaMori

Yep, e to any constant is another constant. Any constant is C. So C+C=C. C*C=C. C^C = C.

ronya

... 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.

Smug Duckling

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.

lunchbox12682

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.

ronya

It probably isn't; writing e^c = C is a common way to reduce the visual complexity of an expression.

Teslan26

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.