Direction Fields

A direction field shows the slope of the DE \(y^{'} = f(x,y)\) at each point \((x,y)\). The figure below shows an example direction field. At each point \((x,y)\), a lineal element is shown depicting the slope. The three solutions below follow the direction of the fields for different particular solutions.

Sketch the direction field for the equation \(\cfrac{dy}{dx} = -2xy\). Sketch the solutions subject to \(y(0)=3\).

Show Answer

First, we can sample points \(x,y\) and plug them into the equation to find the slope:

x-Values	\(0\)	\(1\)	\(1\)	\(2\)	\(2\)	\(2\)
y-Values	\(0\)	\(0\)	\(1\)	\(0\)	\(1\)	\(2\)
dy/dx-Values	\(0\)	\(0\)	\(-2\)	\(0\)	\(-4\)	\(-8\)

At each point, draw a lineal element according to the slope:

Finally, sketch the solution by starting at the initial condition, \(y(0) = 3\). Then, follow the direction fields to sketch the function: