What is the differantial equation which has the particular solution $y_p=(x^2-1)e^{-x}+x$? What is the lowest ordered differantial equation which has the particular solution $y_p=(x^2-1)e^{-x}+x$? 

For $e^{-x}$ we have $(D+1)$
and for $xe^{-x}\to(D+1)^2$, 
For $x^2e^{-x}\to (D+1)^3$

I dont know how to combine them. answer is $(D+1)^3D^2y=0$ btw.
Given: $$y_p = (x^2-1)e^{-x}+x$$
Finding the total derivatives we obtain:
$$dy = [e^{-x}(1+2x-x^2) + 1]dx$$
$$\therefore \frac{dy}{dx} = 1 - e^{-x}(x^2-2x-1)$$
$$\qquad \qquad\qquad\qquad\qquad\qquad\qquad\text{is the required differential equation}  $$
Given: $$y_p = (x^2-1)e^{-x}+x$$
Finding the total derivatives we obtain:
$$dy = [e^{-x}(1+2x-x^2) + 1]dx$$
$$\therefore \frac{dy}{dx} = 1 - e^{-x}(x^2-2x-1)$$
$$\qquad \qquad\qquad\qquad\qquad\qquad\qquad\text{is the required differential equation}  $$