Euler's+Method

= = =** Euler's Method **= The simplest method of obtaining a numerical solution to a differential equation





Ex: if dy = (x + y)dx and f(0) = 1 with a step size of .5, then dy = (0 + 1)(.5) Ex: if y = 1 and dy = .5, then ynew = 1.5 Ex: if x = 0 and the step size (dx) = .5, then xnew = .5 Ex: if you're supposed to find f(1), repeat process until x = 1 Ex: if at x = 1, y = 2, then f(1) = 2 =** REMEMBER TO ALWAYS KEEP TRACK OF WHAT VALUES OF X AND Y YOU ARE PLUGGING IN EACH TIME AND THAT YOUR STEP SIZE (dx) WILL ALWAYS STAY CONSTANT! **=
 * Step 1** - Multiply each side by dx
 * Step 2** - Plug in the starting value of x for x, y for y, and step size for dx into the equation for dy
 * Step 3** - Solve for dy
 * Step 4** - Add the new value of dy to the y value
 * Step 5** - Add the step size to the x value
 * Step 6** - Plug in new values of x and y into the equation for dy
 * Step 7** - Keep track of what values of x and y you are plugging in for each time you solve the equation for dy
 * Step 8** - Repeat steps 2-7 until your x value equals that of the one you were looking for
 * Step 9** - The y value for said x value is your answer

**Example:** f(0) = 6 Stepsize (dx) = .5 f(1) = ?

dy/dx = 2x + 3 dy = (2x + 3y)dx

dy = (2(0) + 3(6))(.5) <-- (0,6) dy = 9

dy = (2(.5) + 3(15))(.5) <-- (.5, 15) dy = 23

dy = (2(1) + 3(38))(.5) <-- (1, 38)

f(1) = 38

__**Video Examples**__

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And if you want the calculator program for Euler's Method:







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