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24 October, 00:34

1.) The independent variable x is missing in the given differential equation. Proceed as in Example 2 and solve the equation by using the substitution u = y'. (y + 7) y'' = (y') 22.) The independent variable x is missing in the given differential equation. Proceed as in Example 2 and solve the equation by using the substitution u = y'. y'' + 6y (y') 3 = 0

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  1. 24 October, 02:44
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    The solution to the differential equation y' (y + 7) y'' = (y') ²

    y = Ae^ (Kx) - 7

    Step-by-step explanation:

    Given the differential equation

    y' (y + 7) y'' = (y') ² ... (1)

    We want to solve using the substitution u = y'.

    Let u = y'

    The u' = y''

    Using these, (1) becomes

    u (y + 7) u' = u²

    u' = u²/u (y + 7)

    u' = u / (y + 7)

    But u' = du/dy

    So

    du/dy = u / (y + 7)

    Separating the variables, we have

    du/u = dy / (y + 7)

    Integrating both sides, we have

    ln|u| = ln|y + 7| + ln|C|

    u = e^ (ln|y + 7| + ln|C|)

    = K (y + 7)

    But u = y' = dy/dx

    dy/dx = K (y + 7)

    Separating the variables, we have

    dy / (y + 7) = Kdx

    Integrating both sides

    ln|y + 7| = Kx + C1

    y + 7 = e^ (Kx + C1) = Ae^ (Kx)

    y = Ae^ (Kx) - 7
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