Q.Verify: $y=ae^x+be^{-x}+x^2$, and $x\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+x^3=xy+2$

$y=ae^x+be^{-x}+x^2$: $\dfrac{dy}{dx}=ae^x-be^{-x}+2x$, and $\dfrac{d^2y}{dx^2}=ae^x+be^{-x}+2=(y-x^2)+2$, i.e. $\dfrac{d^2y}{dx^2}=y-x^2+2$. Multiplying by $x$: $x\dfrac{d^2y}{dx^2}=xy-x^3+2x$ — this is the exact relation $y$ satisfies (checked directly by substitution: both sides equal $x(ae^x+be^{-x})+2x$). Comparing against the stated equation $x\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+x^3=xy+2$: substituting the correct $x\dfrac{d^2y}{dx^2}=xy-x^3+2x$ into it would require the extra term $2\dfrac{dy}{dx}+2x$ to vanish, which it does not in general — so the stated equation carries a spurious $2\,dy/dx$ term (a printing slip …

Differentiate the given relation (implicitly if needed) as many times as there are arbitrary constants, eliminate the constants, and confirm the resulting relation betw …

Trying to isolate y explicitly before differentiating an implicit relation, instead of differentiating implicitly and eliminating the constant directly; sig …