- #1
etotheipi
- Homework Statement
- Show that [itex](\frac{d}{dx} +x)(-\frac{d}{dx} + x)z = -\frac{d^{2}z}{dx^{2}}+x^{2}z + z[/itex]
- Relevant Equations
- N/A
I am wondering why the two methods below give different answers. If I multiply [itex]z[/itex] through the second bracket I get $$(\frac{d}{dx} +x)(-\frac{dz}{dx} + xz)$$which, on expansion, yields $$-\frac{d}{dx}\frac{dz}{dx} -x\frac{dz}{dx} + \frac{d(xz)}{dx} + x^{2}z = -\frac{d^{2}z}{dx^{2}} + x^{2}z + z$$ via the product rule on the third term, as required. However, if instead I expand the brackets first before multiplying through by [itex]z[/itex], I get$$(-\frac{d}{dx}\frac{d}{dx} -x\frac{d}{dx} + x\frac{d}{dx} + x^{2})z = -\frac{d^{2}z}{dx^{2}} + x^{2}z$$ I know the error has something to do with misusing the [itex]\frac{d}{dx}[/itex] operator, but I can't pinpoint it.