The following graph is the member of the family of folia in Problem 51 corresponding to c = 1. Discuss: How can the DE in Problem 51 help in finding points on the graph of x3+ y3= 3xy where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution Ï of the DE? Carry out your ideas, and compare with your estimates of the intervals in Problem 50.
Data from problem 51
The graphs of members of the one-parameter family x3 + y3 = 3cxy are called folia of Descartes. Verify that this family is an implicit solution of the first-orde differential equation
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