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Math Help - First-Order Differential Equations

  1. #1
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    First-Order Differential Equations

    How would you solve this problem? Here is the question:

    Given the differential equation: y' + y tanx= cosx

    (a) Verify by subsitution that for any number C, y(x) = (x+C) cosx satisfies the given differential equation.
    (b) Determine a value of the constant C so that y(3.14)= 0
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  2. #2
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    Quote Originally Posted by googoogaga View Post
    How would you solve this problem? Here is the question:

    Given the differential equation: y' + y tanx= cosx

    (a) Verify by subsitution that for any number C, y(x) = (x+C) cosx satisfies the given differential equation.
    (b) Determine a value of the constant C so that y(3.14)= 0
    Mostly this is just substitution:
    a) y(x) = (x + C)cos(x)

    Thus
    y^{\prime} = cos(x) - (x + C)sin(x)

    Thus the DEq says:
    (cos(x) - (x + C)sin(x)) + (x + C)cos(x) \cdot tan(x) = cos(x)

    cos(x) - (x + C)sin(x) + (x + C)sin(x) = cos(x)

    cos(x) = cos(x)

    So this is true.

    b) We want a value for C such that y(\pi) = 0

    0 = (\pi + C)cos(\pi)

    0 =-(\pi + C)

    C = - \pi

    -Dan
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