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Solve $(D^3-2D^2-5D+6)y=e^{3x}+8$ .
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$$ $$ $ \text{ The Auxillary equation is } \\ $

$ D^3 - 2D^2 -5D + 6 = 0 \\ $

$D= 1,-2,3 \\ $

$ \text{ C.F. is } y = c_1 e^x + c_2 e^{-2x} + c_3 e^{3x} \\ $

$ \text{P.I = } \frac{1}{D^3 - 2D^2 -5D + 6} (e^{3x} + 8) \\ $

$= \frac{1}{D^3 - 2D^2 -5D + 6} e^{3x} + 8 \frac{1}{D^3 - 2D^2 -5D + 6} e^{0x} $

$= \frac{x}{10}e^{3x} + 8 \frac{1}{6} e^{0x} \\ $

$= \frac{x}{10}e^{3x} + \frac{4}{3} \\ $

$\therefore \text{ The complete solution is y = C.F. + P.I. } \\ $

$ \therefore y = c_1 e^x + c_2 e^{-2x} + c_3 e^{3x} + \frac{x}{10}e^{3x} + \frac{4}{3} $

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