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6.1 Differential Equations & Slope Fields

6.1 Differential Equations & Slope Fields. Differential Equations. Any equation involving a derivative is called a differential equation . The solution to a differential is a family of curves that differ by a constant. Example: Find all functions that satisfy .

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6.1 Differential Equations & Slope Fields

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  1. 6.1Differential Equations & Slope Fields

  2. Differential Equations • Any equation involving a derivative is called a differential equation. • The solution to a differential is a family of curves that differ by a constant. • Example: Find all functions that satisfy . • y = x4 – x3 + C • The solution to an initial value problem (a problem involving a differential equation given an initial condition) is a member of the family of curves with a specific constant. • Example: Find the particular solution to the equation whose graph passes through the point (1, 0). • General Solution:y = ex – 2x3 + C when x = 1, y = 0, so 0 = e1 – 2(1)3 + C 2 – e = C Therefore, the particular solution isy = ex – 2x3 + 2 – e

  3. Differential Equations • Example: Find the solution to the differential equation f’(x)= e-x2for which f(7) = 3. • We do not know an antiderivative for f’(x)= e-x2 , so we have to get a little creative with our answer. allows us to find the antiderivative of e-x2 . Allows us to use the Fundamental Theorem to produce the derivative given by the differential equation and satisfy the initial condition.

  4. Slope Fields • Slope fields can help us produce the family of curves that satisfies a differential equation. • Remember: Differential equations give the slope at any point (x, y), and this information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. This process will be repeated for several points to produce a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but recent AP tests have asked students to draw a simple one by hand.

  5. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 0 0 0 0 1 0 0 2 0 0 3 0 2 1 0 1 1 2 2 0 4 -1 -2 0 0 -4 -2

  6. If you know an initial condition, such as (1,-2), you can sketch the particular curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

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