Let f x be a differentiable function such that f ' x = 7 - 3 4 f x x ,   x > 0 and f 1 ≠ 4 .…

Let fx be a differentiable function such that f'x=7-34fxx, x>0 and f1≠4. Then limx→0+⁡x f1x
  1. does not exist.
  2. exists and equals 4.
  3. exists and equals 47.
  4. exists and equals 0.

Solution

We have, dydx+3y4x=7 ( where y=f(x))

This is a linear differential equation.

Integrating factor, I.F. =e∫3 dx4x=e34lnx=x34

So, integrating the given differential equation using I.F. 

⇒y.x34=∫7.x34dx

⇒y.x34=7.4x747+C

⇒yx34=4x74+C

⇒f(x)=4x+Cx-34

⇒limx→0+xf1x⇒limx→0+x4x+Cx34=4

Asked in: JEE Main 2019 (10 Jan Shift 2)

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