Consider the differential equation dy/dx=xy-y. Find d^2y/dx^2 in terms of x and y. Describe the region in the xy plane in which all solutions curves to the differential equation are concave down.

Answer 1

just take d/dx(xy-y)
y + xy' - y'
y+(x-1)y'
y+(x-1)(xy-y)
y+y(x-1)^2
y(1+(x-1)^2)
y" y(x^2-2x+3)

to find y,
dy/dx = xy-y = y(x-1)
dy/y = (x-1)dx
ln y = 1/2 (x-1)^2 + c
y = ce^(1/2 (x-1)^2)
= ce^(1/2x^2 - x + 1/2)
= ce^(1/2 x(x-2))
c changes with rearrangement

y is concave down when y" < 0
since e^(f(x)) i always positive, I don't see anywhere where y is concave down, unlesss c<0