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19 October, 19:58

Find the particular solution that satifies the differential equation and the initial condition. f" (x) = x2 f' (0) = 8, f (0) = 4

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  1. 19 October, 20:10
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    f (x) = x^4/12 + 8x + 4

    Step-by-step explanation:

    f" (x) = x^2

    Integrate both sides with respect to x

    f' (x) = ∫ x^2 dx

    = (x^2+1) / 2+1

    = (x^3) / 3 + C

    f (0) = 8

    Put X = 0

    f' (0) = 0 + C

    8 = 0 + C

    C = 8

    f' (x) = x^3/3 + 8

    Integrate f (x) again with respect to x

    f (x) = ∫ (x^3 / 3) + 8 dx

    = ∫ x^3 / 3 dx + ∫8dx

    = x^ (3+1) / 3 (3+1) + 8x + D

    = x^4/12 + 8x + D

    f (0) = 4

    Put X = 0

    f (0) = 0 + 0 + D

    4 = D

    Therefore

    f (x) = x^4 / 12 + 8x + 4
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