In Example 6.2 (page 222) we found the brachistochrone by exchanging the variabl
ID: 1913723 • Letter: I
Question
In Example 6.2 (page 222) we found the brachistochrone by exchanging the variables x and y. Here is a method that avoids that exchange: Write the time as in Equation (6.19) but using x as the variable of integration. Your integrand should have the form f(y, y', x) = J(y'2 + 1)/y. Since this is independent of x, you can invoke the "first integral" (6.43) of Problem 6.20. Show that this differential equation leads you to the same integral for x as in Equation (6.23) and hence to the same curve as before.
Equation 6.19:
In Example 6.2 (page 222) we found the brachistochrone by exchanging the variables x and y. Here is a method that avoids that exchange: Write the time as in Equation (6.19) but using x as the variable of integration. Your integrand should have the form f(y, y', x) = J(y'2 + 1)/y. Since this is independent of x, you can invoke the first integral (6.43) of Problem 6.20. Show that this differential equation leads you to the same integral for x as in Equation (6.23) and hence to the same curve as before. Equation 6.19:Explanation / Answer
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