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Section 9.1

Section 9.1. Direction Fields. DIRECTION FIELD. A direction field is basically a “flow pattern” used to establish the existence of, and possibly locate an approximate solution curve for a first-order differential equation that cannot be solved by standard techniques.

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Section 9.1

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  1. Section 9.1 Direction Fields

  2. DIRECTION FIELD A direction field is basically a “flow pattern” used to establish the existence of, and possibly locate an approximate solution curve for a first-order differential equation that cannot be solved by standard techniques. For example, the techniques we have learned do not apply to the differential equation

  3. LINEAL ELEMENTSAND ISOCLINE A lineal element is a short line segment, with a given slope (m), and whose midpoint is at a given point (a, b). An isocline is a curve along which the inclination (of the tangents) is the same.

  4. ISOCLINES, DIRECTION FIELDS, AND DIFFERENTIAL EQUATIONS For the differential equation y′ = f (x, y), each member of the one-parameter family of curves f (x, y) = c, c a constant, is an isocline on which we can construct lineal elements, all of which have the same slope. The totality of all lineal elements on all members of the family of curves, f (x, y) = c, is called a direction field, slope field, or lineal element field of the differential equation y′ = f (x, y).

  5. APPROXIMATE SOLUTIONS TO A DIFFERENTIAL EQUATION A solution to the differential equation y′ = f (x, y) that passes through the point (a, b) can be represented, approximately, by a curve through the point (a, b) and through the isoclines with the appropriate slopes.

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