Question

(a) Find fx(x,y), fy(x,y), fx(1,3), and fy(-2,4) for the given function. If 

𝑧 = 𝑓(𝑥, 𝑦) = 3𝑥

ଷ𝑦

ଶ − 𝑥

ଶ𝑦

ଷ + 4𝑥 + 9

Answer 1

Given a constant k ∈ R, find all solutions f : R → R to the differential equation

$f '(x) = k f (x)$

Multiply the equation above $f '(x) − kf (x) = 0\space by \space e^{−kx}$

$f '(x) e^{−kx }− f (x) ke^{−kx} = 0.$

The left-hand side is a total derivative,$[f (x) e^{−kx}0] = 0.$

The solution of the equation above is$f (x)e−kx = c, with c ∈ R. Therefore, f (x) = c ekx.$