The following video goes over the details and shows how to deduce information about the solutions of the corresponding differential equations.
Direction fields are a good way to visualize the solutions of a differential equation.
Since $\displaystyle\frac{dy}{dx} = f(x,y)$, at each point $\left(x_0,y_0\right)$ in the $xy$-plane we can draw a short line with slope $f\left(x_0,y_0\right)$. This is called a direction field. A solution to the differential equation $\displaystyle\frac{dy}{dx} = f(x,y)$ is then a curve that is everywhere tangent to the direction field.
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Examples:
The direction fields of $$\displaystyle\frac{dy}{dx} = x+y,\qquad \displaystyle\frac{dy}{dx} = 2x, \quad\text{ and } \quad \displaystyle\frac{dy}{dx}=y,$$ are (respectively)
The following video goes over the details and shows how to deduce information about the solutions of the corresponding differential equations. |