Differential Equations · Question 37

Let for some function y = f(x), 0x t f(t) dt = x2 f(x), x > 0 and f(2) = 3. Then f(6) is equal to

1. 1
2. 6
3. 2
4. 3

Answer

(1) 1

Solution

Given: 0x t f(t) dt = x2 f(x)

Differentiating both sides w.r.t. x (Leibniz rule):

x f(x) = 2x f(x) + x2 f'(x)

x2 f'(x) = -x f(x)

f'(x)f(x) = - 1x (x > 0)

|f(x)| = - x + C

f(x) = kx

f(2) = 3 k2 = 3 k = 6

f(x) = 6x

f(6) = 66 = 1

Answer: (1)