Hello, mark!
im having trouble with this problem
. . . I'm not surprised!
Obtain
dxdy:ey+ex=sinhy
heres my work
ey+ex=sinhy
ln(ey)+ln(ex)=ln(sinhy) . . . what?
y+x=ln(sinhy) . . . WHAT?
dxdy+1=cothy(dxdy)
from here i do not know what to do
Yes, you do . . . solve for dy/dx . . .
We have: \(\displaystyle \L\,e^y\,+\,e^x\;=\;\sin y\)
Differentiate implicitly: \(\displaystyle \L\,e^y\left(\frac{dy}{dx}\right)\,+\,e^x\;=\;\cosh y\left(\frac{dy}{dx}\right)\)
Okay, one more time . . .
How do we solve for
dxdy ?
We get all the
dxdy terms on one side . . .
\(\displaystyle \L\,e^y\left(\frac{dy}{dx}\right)\,-\,\cosh y\left(\frac{dy}{dx}\right)\;=\;-e^x\)
Factor: \(\displaystyle \L\:\left(e^y\,-\,\cosh y\right)\left(\frac{dy}{dx}\right)\;=\;-e^x\)
Therefore: \(\displaystyle \L\:\frac{dy}{dx}\;=\;\frac{-e^x}{e^y\,-\,\cosh y}\;\;\) . . . or: \(\displaystyle \L\,\frac{e^x}{\cosh y\,-\,e^y}\)