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Math Help - Gradient system

  1. #1
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    Gradient system

    Given that a system is a gradient system, here's how to find it's potential function V. Suppose that x'=f(x,y) and y'=g(x,y). Then x'=-\nabla V implies f(x,y)=-\frac{\partial V}{\partial x} and g(x,y)=-\frac{\partial V}{\partial y}.

    Use this procedure to find V for the following gradient system:
    x'=y^2+y cos x
    y'=2xy+sin x
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  2. #2
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    Quote Originally Posted by splash View Post
    Given that a system is a gradient system, here's how to find it's potential function V. Suppose that x'=f(x,y) and y'=g(x,y). Then x'=-\nabla V implies f(x,y)=-\frac{\partial V}{\partial x} and g(x,y)=-\frac{\partial V}{\partial y}.

    Use this procedure to find V for the following gradient system:
    x'=y^2+y cos x
    y'=2xy+sin x
    Can you state your question differently.
    You want to find a vector field F(x,y) such as what?
    \nabla F=<y^2+y\cos x,2xy+\sin x>
    In that case,
    F_x=y^2+y\cos x
    F_y=2xy+\sin x
    Thus, (in first equation)
    F=xy^2+y\sin x+f(y)
    Where f(y) is to be determined.
    But then,
    F_y=2xy+\sin x+f'(y)
    So, f'(y)=0
    So, f(y)=C
    Thus, it seems the function,
    F(x,y)=xy^2+y\sin x+C
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