Question: A family of curves can often be characterized as the general solution of y' = f(x, y). (a) Show that for the circles with center
A family of curves can often be characterized as the general solution of y' = f(x, y).
(a) Show that for the circles with center at the origin we get y' = -x/y.
(b) Graph some of the hyperbolas xy = c. Find an ODE for them.
(c) Find an ODE for the straight lines through the origin.
(d) You will see that the product of the right sides of the ODEs in (a) and (c) equals -1. Do you recognize this as the condition for the two families to be orthogonal (i.e., to intersect at right angles)? Do your graphs confirm this?
(e) Sketch families of curves of your own choice and find their ODEs. Can every family of curves be given by an ODE?
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