`y = sinx` Determine whether the function is a solution of the differential equation `xy' - 2y = x^3e^x`

First, determine the derivative of the given function:  `y'(x) = cos(x).`  Then substitute `y` and `y'` into the given equation:

`x cos(x) - 2sin(x) = x^3 e^x.`

Is this a true equality for all x? No. To prove this, divide by `x:`

`cos(x) - 2sin(x)/x = x^2 e^x.`

We know that `sin(x)/x` is a bounded function (it is obviously `lt=1` by the absolute value if `|x|gt=1` ), and `cos(x)` is also bounded, but `x^2 e^x` tends to infinity when `x->+oo.` Therefore this equality is false for all x's large enough.