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What Are Integral Curves and Direction Fields Used for?

Integral Curves

Integral curves and family of curves

An integral curve—also known as a parametric curve—is the graph of a particular solution of a differential equation—that is, a solution where the constants are determined.

You can create these curves for particular solutions of both first and second order differential equations. If you have a collection of integral curves with different particular solutions, they are called a family of curves.

Direction Fields

Direction field or slope field

If you have a differential equation of the first order, you can find the tangent at any point (x,y). You do this by moving terms around so that y ends up alone on the left-hand side of the equation. This gives you a formula where you can enter x and y into the formula for y. If you do this for all the points in a region you get a direction field or slope field.

This direction field also shows which direction the integral curve would go for those specific points. You can say how y develops from a point (x,y) by looking at which direction the direction field shows at that point (x,y).

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