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

Options

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

Answer

B

Explanation

`x^(3)(dy)/(dx)+4x^(2)tany=e^(x)secyimpliesx^(3)cosy(dy)/(dx)+4x^(2)siny =e^(x)`
`implies x^(3).(dt)/(dx)+4x^(2).t=e^(x), " "("where " t=siny)`
`impliesx^(4)(dt)/(dx)+4x^(3)t=xe^(x)implies(d)/(dx)(x^(4).t)=xe^(x)impliesx^(4)t=(x-1)e^(x)+c`
`impliesx^(4)siny=(x-1)e^(x)+c`
`becausey(1)=0impliesc=0impliessiny=(e^(x)(x-1))/(x^(4))`