( D^2+D+1)^2(D 2)y=0 ⇒ D=2 #160; #160; #160; D^2+D+1=0 ⇒ D=1±√(1 4)2 #160; #160; #160; #160; #160; = 1±√( 3)2 ...

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Fundamental Laws of Logarithms

Solve D2+D+...

Question

Solve
$(D^{2} + D + 1)^{2} (D - 2) y = 0$

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Solution

${(D^{2} + D + 1)}^{2} (D - 2) y = 0$
$\Rightarrow D = 2$ $D^{2} + D + 1 = 0$
$\Rightarrow D = \frac{1 \pm \sqrt{1 - 4}}{2}$
$= \frac{- 1 \pm \sqrt{- 3}}{2}$
$= \frac{- 1 \pm i \sqrt{3}}{2}$
$\Rightarrow C F = A e^{2 x} + B e^{\frac{- 1 + i \sqrt{3}}{2}} + C e^{\frac{- 1 - i \sqrt{3}}{2}}$
$= A e^{2 x} + e^{- x / 2} (B e^{\sqrt{3} L / 2 x} + C e^{- \sqrt{3} / 2 L x})$
$= A e^{2 x} + e^{- x / 2} (B cos \frac{\sqrt{3}}{2} x + B sin \frac{\sqrt{3}}{2} x + C cos \frac{\sqrt{3}}{2} x - C sin \frac{\sqrt{3}}{2} x)$
$= A e^{2 x} + e^{- x / 2} cos \frac{\sqrt{3}}{2} x (B + C) + e^{- x / 2} sin \frac{\sqrt{3}}{2} x (B - C) .$
Hence, solved.

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