Download Lecture Notes for Section 2.7

ODE Lecture Notes
Section 2.7
Page 1 of 4
Section 2.7: Numerical Approximations: Euler’s Method
Big idea:. If you can’t solve a differential equation explicitly, you can obtain a numerical
approximation to the solution using Euler’s Method, which is a tangent line technique.
Big skill: You should be able to obtain a numerical solution using Euler’s Method by hand and
with a graphing calculator.
Euler’s Method: A technique for obtaining a numerical approximation to points on the curve of
the solution of a differential equation.
We start by thinking about sub-dividing a region of x-values from a to b into n equal
subdivisions, and integrating the differential equation using a left-endpoint numerical
approximation:
y   f  x, y 
y 
y
x
y
 f  x, y 
x
y  f  x, y  x
yi 1  yi  f  xi , yi  x
yi 1  yi  f  xi , yi  x
Or, letting h = x, yi 1  yi  hyi
ODE Lecture Notes
Section 2.7
Page 2 of 4
Practice:
dy
1
 3  2t  y , y(0) = 1 on the interval [0, 1]
dt
2
analytically and using Euler’s method with step sizes of h = 0.2, 0.1, and 0.05. Compare
the results to the answers from the solution function y  t   14  4t 13et /2
1. Solve the initial value problem
ODE Lecture Notes
Section 2.7
Page 3 of 4
ODE Lecture Notes
Section 2.7
Page 4 of 4
To perform Euler’s method quickly on your calculator:
 Store the initial conditions in variables X and Y.
 Store the step size in variable H.
 Enter the update formula for Y, then a colon, then the update formula for X, then a colon,
then the variable Y
 Repeatedly hit the ENTER button until x-value.
 Example: for y  x  e y , y(1) = -2, h = 0.25
1X
-2  Y
.25  H
Y + H*(X + e^(-Y))  Y:X+0.25X:Y
ENTER
ENTER
ENTER …