AP Calculus AB: 11.1.5 Euler's Method for Solving Differential Equations Numerically

• Compute and graph approximate solutions to differential equations using Euler’s Method.

• When the exact solution is known, find the error obtained using Euler’s method with varying step sizes.

• Consider the differential equation y ́ = F(x, y), with initial
  condition y ́(0) = y 0 . It may be impossible to solve this
  differential equation exactly. Using a numerical solution
  procedure called Euler’s method, the solution can be
  approximated by a piecewise linear function.

• Euler’s idea: Start at (x 0 , y 0 ) and follow the tangent line with slope y ́ = F(x 0 , y 0 ) to a new point
  (x 1 , y 1 ) = (x 0 + h, y 0 + hF(x 0 , y 0 ) ). Repeat the process with (x 1 , y 1 ) as the new starting point.

• h is called the step size. It is usually chosen to be small
  because linear approximations to a function at a point are only guaranteed to be accurate nearby that point.

• The x n points increment by the step size h at each step of the process. So x n = x 0 + nh.

• The slope of the function at the point (x n − 1 , y n − 1 ) is F(x n − 1 , y n − 1 ). Starting from this point, the linear
  approximation to the solution moves h units in the x-direction (run), and hF(x n − 1 , y n − 1 ) units in the y-direction (rise). This makes sense, since the slope is rise divided by run, or F(x n − 1 , y n − 1 ).

• Each step of Euler’s Method creates a new point (x n , y n ). The line segment between the current point and the previous point is a linear approximation of the curve.

• Using the direction field for the differential equation, the points obtained via Euler’s Method can be plotted, and the line segments drawn between them to create the piecewise linear approximation to the solution of the differential equation.

• What the solution curve looks like depends a lot on the initial condition, (x 0 , y 0 ).

• Here is another solution curve for the same direction field. Notice that the differential equation is the same, but the initial condition has changed. Now, y(0) = −1.

• Notice how the approximate solution follows the curve indicated by the direction field.

• For these examples, the function that is being substituted for F(x, y) is y − x. Remember that for a different differential equation, this function could be different.

• When you use a smaller step size, you get a better
  approximate solution obtained via Euler’s Method. This
  occurs because the linear approximation obtained at each step is better when the region where the linear approximation is being obtained is smaller.

• To find the approximate value of y(0.2), it will take one step of Euler’s Method with a step size of 0.2, and two steps with a step size of 0.1.

• When the step size is 0.2, the approximate value of y(0.2) is 1.2. When the step size is 0.1, the approximate value of y(0.2) is 1.21.

• Since the exact solution to this differential equation is known, the error can be obtained for each step size by subtracting the approximate solution value from the exact solution value at x = 0.2. With a step size of 0.2, the error is about 0.0214. With a step size of 0.1, the error is about 0.0114.

y(0.6) ≈ 210

0.4683

y(0.6) ≈ 1.9308288

3.3891

y(3) ≈ 7

2.3266

y(2)≈6.5625

y(2) ≈ 3.28125

y(3) ≈ 55

0.7183