) If F(ƒ) = ƒ(§) is the Fourier transform of f(x), show that the Fourier transform of x f(x) is F(xf) = ¸df dε' where is the Fourier transform variable. O Airy's equation is d²u xu = 0. dx² Use Fourier transforms and the result of part (a) to show that, by a suitable choice of a constant of integration, Airy's equation has the Airy function solution 1 Ai(x)= ½ cos (}§³+x) d£. π = COS

Prove the differentiation formula d/dx [coth x] = −csch2 x

FIND THE SOLUTION OF USING EXACT DIFFERENTIATION a) (3x-25²)dx- 2xy dy=0

Find the equation of the tangent line T to the graph of f(x) = 8 sinx at the given point (6,8 sin 6). a. y = (8 sin 6)(x+6) + 8cos 6 b. y = -(8cos 6)(x-6) +8sin 6 c. y=(8cos 6x)(x-6) + 8 sin 6 d. y = -(8cos6)(x-6) + 8sin 6 E. y =(8sin6x)(x+6) + 8cos6