Second derivative of h plus B times h prime plus C times So what does that mean? That means that A times the And that worked out well,īecause, h for homogeneous. Is a solution of the homogeneous equation. Of the homogeneous equation plus a particular This non-homogeneous equation is actually the general solution Before I show you an actualĮxample, I want to show you something interesting. Plus B times the first derivative plus C times theįunction is equal to g of x. So what does all that mean? Well, it means an equation Non-homogeneous second-order linear differential equations Nevertheless, finding even one solution doesn't always turn up to be as easy as it may sound. In the example we could have guessed e^(-x) - 1/2 e^(2x), checked that this is in fact a solution, and used that to write all solutions (by adding the same parametrized homogeneous solution). When looking for a particular solution (solution to the original, inhomogeneous equation) it's enough to find one, no matter which one. The part of the video from 17' to 28' also includes a bit more general demonstration of what Sal shows in his video - that such sums actually are solutions - as well as an explanation of the "particular solution" language. You may find a proof that such sums cover all the possible solutions in the 12th lecture of the 18.03 course. In the inhomogeneous linear case every solution may be expressed as a sum of an arbitrary solution to the inhomogeneous equation plus a solution to the associated homogeneous equation. Homework problems contain all the main types of possible exam questions.Differential equations in general have a whole class of solutions, each making the equality true. Here is a few review problems to make sure you haven't overlooked an important topic:Ĥ.Rev: 3, 4, 6, 7, 9, 11, 13, 22, 25, 26, 32 5.Rev: 2, 3, 5, 6, 11, 12, 17, 18 6.Rev: 3, 4, 7, 8, 15 (find the formula for the coefficients, and write out explicitly the first few terms) 19, 20.
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