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We have a solution for a differential equation which is of the form:

$y(t) \LARGE = \frac{1}{C_1e^t+1}$

Since all the constant does is shift the y(t) to left or right, we can substitute $\large ce^{t} = e^{t+f(C_1)}$ and write the equation as:

$\LARGE\frac{1}{e^{t+f(C_1)}+1}$

How do I solve $f(C_1)$ (in other words how do I go from the first solution to the second one)?

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    $\begingroup$ Do you want to get $f(C_1)$ given $C_1$? That is just $\ln C_1$. $\endgroup$ – KittyL Dec 11 '15 at 18:10
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Solution:

Rewrite C1 as $\large e^{ln (C1)}$

$y(t) \LARGE = \frac{1}{e^{ln(c_1)}e^t+1}$

Rewriting:

$y(t) \LARGE = \frac{1}{e^{t+ln(c_1)}+1}$

Thus:

$f(c_1) = ln(c_1)$

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