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Math Help - Differential equation order3

  1. #1
    Super Member dhiab's Avatar
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    Differential equation order3

    Solve : y''' + y'' + y' = 8(x2+2x+2)
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  2. #2
    MHF Contributor chisigma's Avatar
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    The DE ...

    y^{'''} + y^{''} + y^{'} = 8(x^{2} + 2x + 2) (1)

    ... has 'characteristic polynomial' ...

    t^{3} + t^{2} + t (2)

    ... and its roots are t=0 ,  t= - \frac{1}{2} \pm i \frac{\sqrt{3}}{2}. The 'general integral' of the 'incomplete equation' is then...

    y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x) (3)

    It is easy to verify that y= \frac{8}{3} x^{3} also satisfies (1) so that the general integral of (1) is...

    y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x) + \frac{8}{3} x^{3} (4)

    Kind regards

    \chi \sigma
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