Show that `y=x sin x` is a solution of the differential equation `(d^(2)y)/(Dx^(2))+y-2cosx=0`.

Explanation

`y=x sin x`……….`(1)`
Differentiate with respect to `x`
`(dy)/(dx)=x*cosx+sinx`………..`(2)`
Again differentiate with respect to `x` …………`(2)`
Again differentiate with respect to `x`
`(d^(2)y)/(dx^(2))=x(-sinx)+cosx+cosx`
`=-x sin x+2cosx`
`=-y +2cos x`
[From eq. `(1)`]
`implies (d^(2)y)/(dx^(2))+y-2cosx=0`
`:. y=sinx` is a solution of the given differential equation.