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2.2: FIRST ORDER LINEAR, II This handout describes a much simplified version of the text's
 

Summary: 2.2: FIRST ORDER LINEAR, II
This handout describes a much simplified version of the text's
`Euler-Lagrange Two Stage Method'or `Variation of Parameters'. It
is called the METHOD OF UNDETERMINED COEFFICIENTS. (Sorry for
all the terminology. You will need to know this name; it comes up
again in Math 5A.)
Instead of just telling you an algorithm; I will explain an idea that
ties some stuff we've done so far together. Consider the following
two examples of ODEs:
y + y = t and y + y = 2 sin(t).
If we apply the technique of direction fields we already learned, we
see that neither one has any equilibrium solutions y = constant. For
the first, the isoclines are all straight line parallel to y = t - 1. Above
the line y = t - 1, the slopes are less than 1, and the concavity is
`up'. Below the line y = t - 1, the slopes are all greater than 1 and
the concavity is `down.' See the top of Figure 1 for the computer
generated picture. If that makes you wonder if the isocline y = t - 1
actually is a solution, congratulations. You can easily now check that
y = t - 1 actually is a solution. Furthermore, the other solutions
seems to be drawn towards it as t increases.

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics