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?
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.)