# Need to find the equation of a curve having only the direction of it at a given point

Temperature T of a plate lying in xy plane is defined T(x,y)=50-(x^2)-(2y^2). An ant, which is initially at (2,1) moves along a curve ensuring the temperature is decreasing as rapidly as possible. I need to find the equation of this curve.

The gradient vector is <-2x, -4y>, but I need to go to the decreasing side, therefore the direction is <2x, 4y>.

Having this information, how can I find the equation of the curve?

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Should it not be dy/dx = -(-4y/-2x)? –  CAF Oct 7 '12 at 12:59
Translate the condition to the differential equation $$\frac{dy}{dx}=\frac{-4y}{-2x},$$ which is separable, and easily solved. (Remember to use the initial condition to evaluate the constant of integration.)