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.