18.03 Class 2, February 10, 2006
Numerical Methods
[1] The study of differential equations rests on three legs:
. Analytic, exact, symbolic methods
. Quantitative methods (direction fields, isoclines
....
)
. Numerical methods
Even if we can solve symbolically, the question of computing values
remains.
The number e is the value y(1) of the solution to
y' = y with y(0) = 1. But how do you find that in fact
e = 2.718282828459045
....
? The answer is: numerical methods.
As an example, take the first order ODE y' = x  y^2 = F(x,y)
with initial condition y(0) = 1. Question: what is y(1) ?
I revealed a picture of the direction field with this solution sketched.
This is what we had on Wednesday but upside down: then we considered
y' = y^2  x . The funnel is at the top
this time. This solution seems to be one of those trapped in the funnel,
so for large x , the graph of y(x) is close to the graph of \sqrt
(x) .
But what about y(1) ?
Here's an approach: use the tangent line approximation!
Since F(0,1) = 1 , the straight line best approximating the integral
curve
at (0,1) has slope 1 , and goes through the point (0,1) : so this
gives the estimate y(1) is approximately 0 .
Well, we know that the integral curve is NOT straight. What to do?
Approximate it by a polygon!
So use the tangent line approximation to go half way, and then check
the direction field again:
y(.5) is approximately 1 + (slope)(run) = 1 + (1)(.5) = .5
(.5,.50) is a vertex on this polygon. The direction field there has
slope
F(.5,.5) = .5  (.5)^2 = .25 .
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 Winter '08
 Staff
 Differential Equations, Numerical Analysis, Equations, Euler, direction ﬁeld

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