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2 May, 01:31

The exponential function y (x) = Ceαx satisfies the conditions y (0) = 2 and y (1) = 1. (a) Find the constants C and α. Enter the exact value of α. Enclose arguments of functions, numerators, and denominators in parentheses. For example, sin (2x) or (a-b) / (1+n).

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  1. 2 May, 04:26
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    a = - 0.6931471806

    C = 2

    Step-by-step explanation:

    Given that y (x) = Ce^ (ax) satisfies the conditions y (0) = 2, and y (1) = 1.

    We need to find the values of a and C.

    To do this, we are going to apply the conditions given to the function y (x), this way, we will obtain two equations which can then be solved simultaneously.

    Applying y (0) = 2, put y (x) = 2 and x = 0 in y (x) = Ce^ (ax)

    2 = Ce^ (a * 0)

    C = 2 (because e^0 = 1) ... (1)

    Again, applying y (1) = 1, put y = 1 and x = 1 in y (x) = Ce^ (ax)

    1 = Ce^ (a * 1)

    Ce^a = 1 ... (2)

    From (1), C = 2. Putting this in (2), we have

    2e^a = 1

    Divide both sides by 2

    e^a = 1/2

    This equation has an index which could make finding a prove difficult by a direct mean, to find a, we need to take logarithm of both sides,

    log (e^a) = log (1/2)

    (a) log (e) = log (1/2)

    a = log (1/2) / (log e)

    a = (-0.3010299957) / (0.4342944819)

    a = - 0.6931471806
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