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. Collections: Mathematics