Calculus of a Single Variable, Chapter 6, 6.3, Section 6.3, Problem 27

To find the equation of the graph passing through the point (9,1), we need to solve the given differential equation:
y' = y/(2x) .
First, rewrite it as
(dy)/(dx) = (y)/(2x) . This equation can be solved by the method of separating variables.
Multiply by dx and divide by y:
(dy)/y = (dx)/(2x) . Now we can integrate both sides:
lny = 1/2lnx+C = lnx^(1/2)+C , where C is an arbitrary constant.
Rewriting this in exponential form results in
y = e^(ln(x^(1/2)) + C) = e^C*x^(1/2) .
Since the graph of this equation passes through the point (9,1), we can find C:
1 = e^C*9^(1/2)
e^C = 1/3
C = ln(1/3) = -ln3 .
So the equation of the graph passing through the point (9,1) with the given slope is
y(x) = 1/3x^(1/2) = 1/3sqrt(x) .