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Dividing a Monomial by a Monomial

Solve: D4 -...

Question

Solve: $(D^{4} - 4 D^{3} + 6 D^{2} - 4 D + 1) y = 0$ .where $D = \frac{d y}{d x}$

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Solution

$(D^{4} - 4 D^{3} + 6 D^{2} - 4 D + 1) y = 0$
$(D + 1 Q)^{4} y = 0$
Auxiliary equation is $(m + 1)^{4} = 0$
$m = - 1, - 1, - 1, - 1$
Therefore,
$y = C_{1} + C_{2} x e^{- x} + C_{3} x^{2} e^{- x} + C_{4} x^{3} e^{- x}$

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