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The relationship between differential equations and their vector fields:

From Wikipedia:

	a vector field is an assignment of a vector to each point in a
	subset of Euclidean space. A vector field in the plane, for instance,
	can be visualized as a collection of arrows with a given magnitude
	and a direction, each attached to a point in the plane.

attempted translation to basic english:

	every point in the plane has a location. let's call this location (x,y).

	the differential equation tells us where this point wants to move to:

		if dx = 1, it means the point wants
			to move towards the right at a speed of 1.
		if dx = -2, it means the point wants
			to move towards the left at a speed of 2.
		if dy = 3, it means the point wants
			to move up at a speed of 3.
		if dy = -4, it means the point wants
			to move down at a speed of 4.

		if dx = 1 and dy = 1, it means the point wants
			to move to the right at a speed of 1 and up at a speed of 1,
			so overall it will want to move up and to the right
			at a speed of sqrt(1^2 + 1^2) = sqrt(2).


	if both the dx and dy parts of the equation are just numbers, 
	then the vector field is called a CONSTANT vector field.

	if we have a contant vector field with components
	
		dx/dt = 3
		dy/dt = 4
	
	and we drop a ball at location (0,0), it will want to move to the right by 3 units and up by 4 units, so it will move up and to the right at an angle of 37 degrees and a speed of 5 (it's a 3,4,5 triangle)

	In fact, if we drop a ball anywhere in the field it will want to move in the exact same way because this is a constant vector field, and constant vector fields affect everything in the field equally regardless of its position in the field