# Thread: differential functions

1. ## differential functions

I am trying to do this problem using a somewhat similiar book example...

dy/dx = (e^y + 1)(x-2)^9
d/dy (e^y+1) = d/dx (x-2)^9
integral d/dy (e^y +1) = integral d/dx (x-2)^9
ln[y+1]^10 = 9(x-2)^8

2. Originally Posted by startingover
I am trying to do this problem using a somewhat similiar book example...

dy/dx = (e^y + 1)(x-2)^9
d/dy (e^y+1) = d/dx (x-2)^9
integral d/dy (e^y +1) = integral d/dx (x-2)^9
ln[y+1]^10 = 9(x-2)^8
$y' = (e^y + 1)(x - 2)^9$ .............divide both sides by $e^y + 1$

$\Rightarrow \frac {y'}{e^y + 1} = (x - 2)^9$ ........now split up the y' and set up the integration

$\Rightarrow \int \frac {1}{e^y + 1}dy = \int (x - 2)^9 dx$ ........as it stands, the left hand side is hard to integrate

$\Rightarrow \int \frac {e^y + 1 - e^y}{e^y + 1} dy = \int (x - 2)^9dx$ ......add $e^y - e^y$ (which is zero so we're not changing anything) to the numerator on the left.

$\Rightarrow \int 1dy - \int \frac {e^y}{e^y + 1}dy = \int (x - 2)^9dx$ ........split up the integral on the left

$\Rightarrow y - \ln \left|e^y + 1 \right| = \frac {1}{10}(x - 2)^{10} + C$ .....integrated, using substituion on the second integral on the left

if you have any questions say so

3. Originally Posted by startingover
I am trying to do this problem using a somewhat similiar book example...

dy/dx = (e^y + 1)(x-2)^9
We have,
$y' = (e^y+1)(x-2)^9$
Divide,
$\frac{y'}{e^y+1} = (x - 2)^9$
Integrate,
$\int \frac{y'}{e^y+1} dx = \int (x -2)^9 dx$
Substitution rule,
$\int \frac{1}{e^y+1} dy = \int (x-2)^9 dx$
Now you need to find,
$\int \frac{1}{e^y+1} dy = \int \frac{e^y}{e^y(e^y+1)}dy$
Let $t=e^y \mbox{ with }t'=e^y$ thus,
$\int \frac{1}{t(t+1)} dt = \int \frac{1}{t} - \frac{1}{t+1} dt = \ln |t| - \ln |t+1| +C= \ln \left| \frac{t}{t+1} \right| +C = \ln \frac{e^y}{e^y+1} +C =$ $- \ln (1+e^{-y})+C$
Thus, we find that,
$- \ln (1+e^{-y}) = \frac{1}{10}(x-2)^{10} +C$