Solution of differential equation `x^(2)y - x^(3) (dy)/(dx)=y^(4) cos x` is

Options

A. `x^(2)y^(-3) = 2 sin x + c`
B. `x^(2)y^(-3) = 3 cos x + c`
C. `x^(3)y^(-3) = 3 sin x + c`
D. `x^(2)y^(3) = 3 sin x + c x^(2) y `

Answer

C

Explanation

`x^(2) y - x^(3)(dy)/(dx)=y^(4)cos x`
`(1)/(y^(4))(dy)/(dx)+((-1)/(x))(1)/(y^(3))=(cos x)/(x^(3))`
put `(1)/(y^(3))=v`
`(1)/(y^(4))(dy)/(dx)=-(1)/(3)(dv)/(dx)`
`(-1)/(3)(dv)/(dx)+(-(-1)/(x))v=-(cos x)/(x^(3))`
`(dv)/(dx)+(3)/(x)v=3 cos x`
`I.F.=e^(3 int (1/2dx)=x^(3)`
solution `v*x^(3)=3int cos x dx + c`
`(1)/(y^(3))x^(3)=3 sin x + c`
`x6(3)y^(-3)=3 sin x + c`