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Math Help - Tough Differential Equation Problem

  1. #1
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    Tough Differential Equation Problem

    I need to find a function F such that it is continuous everywhere and

    y'(t)=F(y(t)) and y(0)=0

    The only thing i could think of is y(t)=e^t but that obviously doesnt satisfy the initial value. Any help or hints is greatly appreciated.
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  2. #2
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    Re: Tough Differential Equation Problem

    So you're supposed to find both F and y(t) for which the given conditions are true, or ...?

    How about y(t)=\sin t. Then y(0)=\sin 0=0.
    And also y'(t)=(\sin t)'=\cos t =\sqrt{1-\sin^2t}=\sqrt{1-(y(t))^2}=F(y(t)).
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  3. #3
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    Re: Tough Differential Equation Problem

    I forgot to mention that I need to find a function F such that the initial value problem has infinitely many solutions. I don't know if that changes anything....
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  4. #4
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    Re: Tough Differential Equation Problem

    So to clear things up: I don't need to find a function y since this function F should take any function y and spit out its derivative i think.
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  5. #5
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    Re: Tough Differential Equation Problem

    The problem with this question is that the complexity of any derivative is arbitrary, for example, if we know that the function y is of the general form:



    ... where a and b are constants. It is easy to create a function F that will reliably produce y's derivative, as in:



    However, even a slight change in the general form of y will upset everything and F will no longer function, while, as stated earlier the above function F produces the derivative of y, it will not produce the derivative of x (with constants a, b and c):






    However, a bit of manipulation will create a function that will work for both x and y, (above) function G, since they are quite similar. this won't work in all instances; consider the functions p and q, defined below:




    Creating a function that will produce the derivatives of both these functions is an enormous, if not impossible, task. So, my conclusion is, it is possible to create a function F that will produce the derivative of a function y, but the general form of y must first be known. Otherwise we'd have to consider an infinite number of combinations of roots, fractions, logarithms, trigonometric functions and lord knows what else - which from my humble high-school perspective - seems impossible.

    Sorry. =( I hope someone else has a more positive answer.
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