Math 152 - Lecture Notes # 3 - David Maslanka THE CATENARY A ...
Math 152 - Lecture Notes # 3 - David Maslanka THE CATENARY A ...
Math 152 - Lecture Notes # 3 - David Maslanka THE CATENARY A ...
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Problem _______________________________________________________________________________________<br />
Show that y = a cosh ( x a ) + C satisfies the differential equation ( * ) with a = H δ<br />
by doing the following:<br />
1. Substitute z = dy<br />
dx<br />
and<br />
dz<br />
dx = d2 y<br />
dx 2 in ( * ) and obtain a first order DE with dependent variable z .<br />
2. Solve the first order DE in z .<br />
Hint: This involves verifying that:<br />
Then note that<br />
⌠<br />
⎮<br />
⌡<br />
1<br />
d<br />
1 + z 2 z = ln ( z + 1 + z z ) + C .<br />
ln ( z + 1 + z z ) = sinh -1 ( z ) , the inverse hyperbolic sine of z .<br />
3. Replace z in terms of y and solve the resulting first order DE for y .
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