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Math Help - Writing two Second-Order equations as a System of First-Order Equations

  1. #1
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    Writing two Second-Order equations as a System of First-Order Equations

    I'm just looking over some stuff as revision because I have exams soon and I'm a bit confused about this question.

    It says:
    Consider a tennis ball that is thrown upwards at time t = 0, with an initial speed U at an angle theta to the horizontal. From Newton's second law, the equations of motion (neglecting air resistance) can be written as follows:

    (d^2)x/d(t^2) = 0
    (d^2)y/d(t^2) = -g

    Rewrite each of the above second-order differential equations as 2 first-order differential equations.

    No matter how many different ways I try it, I end up with only one first-order differential equation for each of these equations and can't find a second one. Essentially, I always get dx/dt = c, and dy/dt = gt + c. I'm probably being really stupid here, but please help.
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  2. #2
    A Plied Mathematician
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    You have to do a substitution. You're integrating, or solving the equation, which is, of course, the final goal. But this problem is just an exercise in generating first-order systems of equations from higher-order equations - something that definitely has practical value for numerical work. For the first equation, let

    x_{1}=x,\quad x_{2}=\dot{x}.

    Then write two equations that are the equivalent of \ddot{x}=0. Hint: x_{2}=\dot{x} is already one of them.

    The y equations will work in much the same way. What do you get?
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