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Warm up

Warm up. Solve for y if dy/dx = (xy) 2 and y = 1 when x = 1. Find the particular solution of the differential equation below that satisfies the initial condition y( 0) = 7. Euler’s Method and Slope Fields.

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Warm up

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  1. Warm up Solve for y if dy/dx = (xy)2 and y = 1 when x = 1. Find the particular solution of the differential equation below that satisfies the initial condition y(0) = 7.

  2. Euler’s Method and Slope Fields

  3. are a tool to graphically obtain the solutions to a first order differential equation. Slope Fields

  4. The slope field for a certain differential equation is shown. Which of the following could be a particular solution to that differential equation? • y = x2 • y = ex • y = e –x • y = cos x • y = ln x

  5. Given: y’= -2xy Create the slope field. Sketch the particular solution through the point (1, -1). Solve the differential equation to find the specific solution with initial condition y(1) = -1

  6. Which of the following differential equations matches the slope field given?

  7. Which of the following differential equations matches the slope field given?

  8. Leonhard Euler (1707-1783) was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory. He was arguably the greatest mathematician of the eighteenth century (His closest competitor for that title is Lagrange). Euler’s Method …is used to approximate values on the solution graph to a differential equation when you can’t actually find the specific solution to the differential equation using separation of variables.

  9. Let’s start with a differential equation you can solve… Given: Find f(0.8) using Δx = 0.2 and f(0) = 4

  10. f(0) = 4 is our reference point so that is where we start. • Plug the reference point into dy/dx to get the slope of the tangent line at that point. 2. Plug Δx into dx (the run of the slope) dy/dx = slope from #1 3. Solve for dy to get the rise 4. Add dy to the previous y to get the new y . 5. Add dx to the previous x to get the new x . 6. Write the new point. (reference point) 7. Repeat the steps until you can answer the question . 4.112

  11. Use Euler’s Method with Δx = 1/3 to approximate y(3) ifthe point (2,1) appears on the solution graph and 148/27 or 5.481

  12. Use Euler’s Method with Δx = 0.1 to approximate f(3.4) ifthe point (3,4) appears on the solution graph and 3.6751

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