a numerical technique for building a solution to a de or system of de’s
TRANSCRIPT
Euler’s MethodA Numerical Technique for Building a Solution to a DE or
system of DE’s
Slope Fields This is the slope field for
.2 1 .20
dP PP
dt
We get an approx. graph for a solution by starting at an initial point and following the arrows.
Euler’s Method
Dt
We can also accomplish this by explicitly computing the values at these points.
Here’s how it works.
. . . then we project a small distance along the tangent line to compute the next point, . . .
We start with a point on our solution. . .
. . . and repeat!
. . . and a fixed small step size Dt.
Projecting Along a Little Arrow
Dt0 0( , )t y
1 1( , )t y
Dy
Projecting Along a Little Arrow
Dt
0 0( , )dy
f t ydt
0 0( , )t y
slope
1 1( , )t y
= f (t0,y0) Dt
Dy = slope Dt
Projecting Along a Little Arrow
Dt
0 0( , )dy
f t ydt
0 0( , )t y
slope
1 1( , )t y
= (t0 +Dt , y0+ Dy)
= (t0 +Dt , y0+f (t0,y0) Dt)
Projecting Along a Little Arrow
old oldslope ( , )f t y
old old( , )t y
new new( , )t y
= (told +Dt , yold+ Dy)
= (told +Dt , y0+ f (told , yold) Dt)
Summarizing Euler’s Method
and a fixed step size Dt.
an initial condition (t0,y0), A smaller step size will lead to more accuracy, but will also take more computations.
You need a differential equation of the form , ( , )dy
f t ydt=
tnew = told + Dtynew = y0+ f (told , yold) Dt
For instance, if
and (1,1) lies on the graph of y, then 1000 steps of length .01 yield the following graph of the function y.
This graph is the anti-derivative of sin(t 2); a function which has no elementary formula!
2sin( )dy
tdt=
Exercise
Start with the differential equation , the initial
condition , and a step size of Dt = 0.5.
2dyt y
dt=
( )0 0, (2,1)t y =
Compute the next two (Euler) points on the graph of thesolution function.
Exercise
Start with the differential equation , the initial
condition , and a step size of Dt = 0.5.
2dyt y
dt=
( )0 0, (2,1)t y =
0 0
1 1
2 2
( , ) (2 ,1)
( , ) (2 .5 ,1 2(.5)) (2.5 , 2)
( , ) (2.5 .5 , 2 10(.5)) (3 , 7)
t y
t y
t y
=
= + + =
= + + =