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 \( \frac{dy}{dx}=-2xy \). Sketch the solutions subject to \(y(0)=3\).

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Sample points \(x,y\) and plug into the equation to find the slope.

\(x\)	\(y\)	\(\frac{dy}{dx}\)
\(0\)	\(0\)	\(0\)
\(1\)	\(0\)	\(0\)
\(1\)	\(1\)	\(-2\)
\(2\)	\(0\)	\(0\)
\(2\)	\(1\)	\(-4\)
\(2\)	\(2\)	\(-8\)

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.