For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.(i) `y=a e^x+b e^(-x)+x^2` : `x(d^2y)/(dx^2)+2y(dy)/(dx)-x y+x^2-2=0`

Explanation

To verify that the given function \( y = a e^x + b e^{-x} + x^2 \) is a solution of the differential equation

\[ x \frac{d^2y}{dx^2} + 2y \frac{dy}{dx} - xy + x^2 - 2 = 0, \]

we will follow these steps:

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