a numerical technique for building a solution to a de or system of de’s
TRANSCRIPT
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Euler’s MethodA Numerical Technique for Building a Solution to a DE or
system of DE’s
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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.
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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.
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Projecting Along a Little Arrow
Dt0 0( , )t y
1 1( , )t y
Dy
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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
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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)
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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)
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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
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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=
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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.
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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
=
= + + =
= + + =