The DE is: dy/dx – y^2 + 6y – 13 = 0
dy/dx = y^2 – 6y + 13
dy/dx = (y – 3)^2 + 4
dy/dx = v^2 + 4, where v = y – 3, and dv = dy, giving,
dv/dx – v^2 + 4
dv/(v^2 + 4) = dx
Integrating,
(1/2)arctan(v/2) = x + c
arctan(v/2) = 2(x + c)
v = 2.tan(2(x + c))
y – 3 = 2.tan(2(x + c))
y(x) = 3 + 2.tan(2(x + c))