The figure (which is not pictured, sorry) shows the slope field for y =.2(9-y^2). Make a sketch of the solution using the initial value y(-4)=-2. Find the function f(x) that this solution curve is approaching as x--infinity. I need help with finding the function f(x) that the solution curve is approaching as x--infinity. Thank you!

Question

The figure (which is not pictured, sorry) shows the slope field for y =.2(9-y^2). Make a sketch of the solution using the initial value y(-4)=-2. Find the function f(x) that this solution curve is approaching as x-->infinity. I need help with finding the function f(x) that the solution curve is approaching as x-->infinity. Thank you!

Steve · Answer

Assuming you mean

y = .2(9-x^2)

Clearly, y(-4) ≠ -2

So, let's work with something like

y' = .2(9-x^2)
Then there will be families of arrows following trajectories of parabolas.

You can find a nice online site at

https://www.desmos.com/calculator/p7vd3cdmei

Just type in your function and then pick the one where y(-4) = -2.