X y   xy   x  v y  (i) Find regular singular points. (ii) Find irregular singular points. (iii) Suppose infinite series solution about regular singular points. (iv) Find indicial equation. (v) Find indicial roots. (vi) Simplify equation by shifting of summation index. (vii) Find recurrence relation. (viii) Find successive coefficients. (ix) Find general solution.

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: