Transcript sect. 5-J

EULER'S METHOD
Section 5-J
Definition
y '  F ( x, y )
Euler’s Aproximation
Definition
• The smaller the steps size the better the
approximation of the exact solution
Start by finding the tangent line
through A0
1) Use Euler’s Method to approximate the
particular solution of the differential equation
passing through the point (0,1) starting at zero with
five steps of equal width if y '  x  y
1) graph points
Error  exact  approximate
When f(x) is concave
down the estimate is
greater than the exact
solution
When f(x) is concave up
the estimate is less than
the exact solution
2)
Consider the differential equation
dy
 1 y
dx
let y  f (x) be the particular solution to
this differential equation with the initial condition f (1)  0
For this particular solution, f ( x)  1 for all values of x
a) Use Euler’s method, starting
at x = 1 with two steps of equal
size to approximate f(0). Show
the work that leads to your
answer.
2)
Consider the differential equation
dy
 1 y
dx
let y  f (x) be the particular solution to
this differential equation with the initial condition f (1)  0
For this particular solution, f ( x)  1 for all values of x
f ( x)
b) Find lim 3
x 1 x  1
2) Consider the differential equation dy  1  y
dx
c) Find the particular solution to y  f (x)
for the differential equation
dy
 1 y
dx
with the initial conditions f (1)  0
d. Graph and compare the points and the
actual solution. From part A the points are
(1, 0) & (1/2, -1/2) & (0, -5/4)
e. Find the error at f(0)
HOME WORK
Euler’s Method
Worksheet 5-J