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Numerade Educator



Problem 52 Hard Difficulty

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve $ y = f(x) $ that satisfies the differential equation
$ \frac {d^2 y}{dx^2} = \frac {pg}{T} \sqrt {1 + (\frac {dy}{dx})^2} $
where $ p $ is the linear density of the cable, $ g $ is the acceleration due to gravity, $ T $ is the tension in the cable at its lowest point, and the coordinates system is chosen appropriately. Verify that the function
$ y = f(x) = \frac {T}{pg} \cosh (\frac {pgx}{T}) $
is a solution of this differential equation.


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Video Transcript

Why prime we know is gonna be t over. Sign age. Just remember that you're doing the chain rule. You just remember to cancel out what you can. We have signed h of Nation Axe T now the second derivative is gonna be co sign age times PG over tea times PG axe over tea. Therefore, remember that we know the identity co signed h scored Axe is one plus sign each scored X. Therefore we can write. This is PG over tea Cosan Age of PG acts over tea is PG over Tea co signed h PG Axe over Chief