What is expand (D^2-D+3)[x^3e^{-2x}] ?

The solution to expand (D^2-D+3)[x^3e^{-2x}] is D^2e^{-2x}x^3-De^{-2x}x^3+3e^{-2x}x^3

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expand (D^2-D+3)[x^3e^{-2x}]

expand

(D^{2} - D + 3) [x^{3} e^{- 2 x}]

D^{2} e^{- 2 x} x^{3} - D e^{- 2 x} x^{3} + 3 e^{- 2 x} x^{3}

Solution steps

(D^{2} - D + 3) [x^{3} e^{- 2 x}]

= (D^{2} - D + 3) [e^{- 2 x} x^{3}]

Distribute parentheses

= [x^{3} e^{- 2 x}] D^{2} + [x^{3} e^{- 2 x}] (- D) + [x^{3} e^{- 2 x}] \cdot 3

Apply minus-plus rules

+ (- a) = - a

= D^{2} [e^{- 2 x} x^{3}] - D [e^{- 2 x} x^{3}] + 3 [e^{- 2 x} x^{3}]

Remove parentheses:

(a) = a

= D^{2} e^{- 2 x} x^{3} - D e^{- 2 x} x^{3} + 3 e^{- 2 x} x^{3}

\frac{1}{1 - 2 x}

\frac{4 x ^{2} - 1}{( x + 2 ) ^{2}}

\frac{x ( x - 2 )}{( x - 1 ) ^{2}}

- \frac{p}{( p + 1 ) ( p + 2 )}

20000 (x - 4) e^{- \frac{x}{2} - 1}

What is expand (D^2-D+3)[x^3e^{-2x}] ?
The solution to expand (D^2-D+3)[x^3e^{-2x}] is D^2e^{-2x}x^3-De^{-2x}x^3+3e^{-2x}x^3