The solution of the differential equation `x^(3)(dy)/(dx)+4x^(2) tany=e^(x) secy` satisfying `y(1)=0`, is

Options

A. `tany=(x-2)e^(x)logx`
B. `siny=e^(x)(x-1)x^(-4)`
C. `tany=(x-1)e^(x)x^(-3)`
D. `siny=e^(x)(x-1)x^(-3)`

Answer

B

Explanation

We have,
`x^(3)(dy)/(dx)+4x^(2) tan y=e^(x) sec y`
`rArr" "x^(3) cos y (dy)/(dx)+4x^(2) sin y=e^(x)`
`rArr" "x^(4)cos ydy +4x^(3) sin y dx=xe^(x)dx`
`rArr" "d(x^(4)sin y)xe^(x)dx`
On integrating, we get
`x^(4)siny=(-1)e^(x)+C`
It is given that y = 0 where x = 1.
Putting x = 1 and y = 0 in (i), we get C = 0
Putting C = 0 in (i), we get
`x^(4)sin y=(x-1)e^(x)rArr sin y=e^(x)(x-1)x^(-4)`