Determine whether the given function is a solution to the given differential equation.

i$.,x^{\text{'}\text{'}}+x=0,$ with possible solution $x(t)=2cost-3sint$

ii$.,\frac{d^{2}y}{d{x}^2}+y=x^2+x+1,$ with possible solution $y(x)=sinx+x^2+x+1$

iii. $\frac{d^2}{d{t}^2}-\frac{d}{dt}+3=-2e^{2t},$ with possible solution $(t)=2e^{3t}-e^{2t}$

iv$.y^{\text{'}\text{'}}+4y=-5e^{-x},$ with possible solution $y(x)=3sin2x-e^{-x}$