Solution of the differential equation `(dy)/(dx)+ytanx=x^(n)cosx` is

Options

A. `(m+1)y=x^(m+1) cos x + C (m+1) cos x `
B. `my= (x^(m) +C) cos x`
C. `y= (x^(m+1) +C) cos x`
D. None of the above

Answer

a

Explanation

This is the linear differetial equation of the form
` (dy)/(dx) +Py = Q`
where, P = tan x and Q = `x^(m) cos x`
Now , integrating factor (IF) ` e^(intPdx)=e^(inttandx)`
` = e^(log sec x)=sec x `
This solution is given by ,
` y *e ^(int Pdx) = int Q*e^(int Pdx)dx+C`
` rArr y * sec x = int x^(m) cos x * sec x dx +C`
` rArr y sec x = (x^(m+1))/(m+1)+C`
` :. (m+1) y = x^(m+1) cos x +C (m+1) cos x`