The equation x2 + y2 = 2cx defines the family of circles tangent to the y-axis at the orig...

The equation x2 + y2 = 2cx defines the family of circles tangent to the y-axis at the origin.

(a) Show that the family of curves orthogonal to this family satisfies the differential equation

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(b) Find the orthogonal family and provide a sketch depicting the orthogonality of the two families.

Knowing an integrating factor exists and finding one suitable for a particular equation are two completely different things. Indeed, as stated previously, finding an integrating factor can be a genuine mathematical art. However, certain differential forms can remind us of differentiation techniques that may aid in the solution of the equation at hand. For example, seeing x dy + y dx reminds us of the product rule, as in d(xy) = x dx + y dy, and x dy –y dx might remind us of the quotient rule, d(x/y) = (y dx – x dy)/y2. In the equation

we are again reminded of the product rule. In fact, if you multiply the equation by 1 /(xy), then

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