Find family of curves orthogonal to f(x)

I don’t understand what it wants me to do here? By orthogonal does it mean integral( f(x) * cx^2)) = 0 along some limits ( but I don’t know what those limits would be ?)

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Reply 1

Skiwi

Orthogonal just means perpendicular. You want to find a set of curves that are always perpendicular to the set of curves you've sketched in the first bit.

I assume you've done the sketching bit? You should get a decent idea what form the orthogonal curves.

To do this, think about what that means. At any point where the two curves intersect, the derivative of one is the negative reciprocal of the other.
You don't know what the second curve looks like, but you know what it's first derivative must be.
Try finding the first derivative of the curve you're given, but removing the constant. And then solving the differential(making sure to use the fact the first derivative of the orthogonal curve is the negative reciprocal) to find the equation of the orthogonal curves. As you're finding the family of curves, don't worry about limits. You can just merge that all into one constant.

Reply 2

ghostwalker

Original post by grhas98

I don’t understand what it wants me to do here? By orthogonal does it mean integral( f(x) * cx^2)) = 0 along some limits ( but I don’t know what those limits would be ?)

You might find this example useful: