# Thread: Simple but conceptual question

1. ## Simple but conceptual question

So I may just be rusty on my algebra, but I'm covering differential equations at a basic level right now and I've seen this pop up a few times and I'd love to know the proof behind why.

I'll see either $\displaystyle 10-x=e^{-kt-c1}$ become $\displaystyle x=10-Ce^{-kt}$ or say $\displaystyle y+4=e^{\frac{-x^2}{2}+C1}$ become $\displaystyle y+4=Ce^{\frac{-x^2}{2}}$

I just don't quite see how they're pulling that C down and in front of the exponential function...can anyone offer a proof or other explanation as to why? Thanks!

2. In the equations you're giving, they are using little c's... The big C stands for:

$\displaystyle C = \pm e^{c_1}$

Or e to the power of some constant... Since that does not change it's constancy, it's valid.

In most basic differential equations, the C becomes the initial value anyway.

For example:

$\displaystyle 10 - x = e^{-kt - c_1}$

$\displaystyle 10 - x = e^{-kt}e^{-c_1}$

$\displaystyle C = e^{-c_1}$

$\displaystyle 10 - x = Ce^{-kt}$

$\displaystyle 10 - Ce^{-kt} = x$