Show that `x y=a e^x+b e^(-x)+x^2` is a solution of the differential equation `x(d^2y)/(dx^2)+2dy/dx-x y+x^2-2=0.`

Explanation

`x y=a e^{x}+b e^{-x}+x^{2}`


Differentiate the equation with respect to x,


`x frac{d y}{d x}+y=a e^{x}-b e^{-x}+2 x`


Again differentiate the equation with respect to x,


`x frac{d^{2} y}{d x^{2}}+frac{d y}{d x}+frac{d y}{d x}=a e^{x}+b e^{-x}+2`


`x frac{d^{2} y}{d x^{2}}+2 frac{d y}{d x}=a e^{x}+b e^{-x}+2`


Hence,


`x frac{d^{2} y}{d x^{2}}+2 frac{d y}{d x}-x y+x^{2}-2=(a e^{x}+b e^{-x}+2)-(a e^{x}+b e^{-x}+x^{2})+x^{2}-2=0`


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