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Question:

Let f be a differentiable function such that f'(x) = 7f(x)/x, (x > 0) and f(1) ≠ 4. Then limx→0+ x f(1/x):

Exists and equals 4

Exist and equals

Does not exist

Exists and equals 47

Solution:

f'(x) = 7f(x)/x (x > 0) Given
f(1) ≠ 4
limx→0+ x f(1/x) = ?
dy/dx + 34y/x = 7 (This is LDE)
IF = e∫34/x dx = e34ln|x| = x34
y.x34 = ∫7.x34 dx
y.x34 = 7.x74/74 + C
f(x) = 4x + C.x-34
f(1/x) = 4x-1 + C.x34
limx→0+ x f(1/x) = limx→0+ (4 + C.x74) = 4