I was recently asked about how to spot which direction field corresponds to which differential equation. I hope that by working through a few examples here we will get a reasonable intuition as to how to do this.
Remember that a direction field is a method for getting the general behaviour of a first order differential equation. Given an equation of the form:
For any function of x and y, the solution to this differential equation must be some function (or indeed family of functions) where the gradient of the function satisfies the above relationship.
The first such equation that we looked at was the equation:
We are trying to find some function, or indeed family of functions y(x) which satisfy this equation. We need to find a function whose derivative (y'(x)) at each point x is equal to the value of the function (ie. y(x)), plus that value of x.…