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Section 6.1: Euler’s Method

Section 6.1: Euler’s Method. Local Linearity and Differential Equations.

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Section 6.1: Euler’s Method

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  1. Section 6.1: Euler’s Method

  2. Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.) Not a good approximation. Consider smaller pieces. Slope at (2,0): Tangent line at (2,0): Tangent line approximation at x=2.2:

  3. Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.) Slope at (2.2,0.4): “Tangent line” at (2.2,0.4): “Tangent line” approximation at x=2.4:

  4. Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.) Slope at (2.4,0.92): “Tangent line” at (2.4,0.92): “Tangent line” approximation at x=2.6:

  5. Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.) Slope at (2.6,1.584): “Tangent line” at (2.6,1.584): “Tangent line” approximation at x=2.8:

  6. Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.) Slope at (2.8,2.421): We can generalize this process. “Tangent line” at (2.8,2.421): “Tangent line” approximation at x=3: Derivative Previous y Change in x Next y

  7. Euler’s Method Numerically approximate values for the solution of the initial-value problem , , with step size , at , are Change in x Next Approximate Solution Previous Approximate Solution Value of Differential Equation at Previous Point

  8. Example If and if when , use Euler’s Method with five equal steps to approximate when . Euler’s Method 5

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