M427K Handout: Non-homogeneous Second Order DifferentialEquations: Method of Undetermined Coefficients and Variation ofParametersSalman ButtFebruary 21, 2006This handout discusses two techniques to solve second order differential equations that arenon-homogeneous. First some notes:Notes1. If you have not gotten your midterm back yet, see me after class or in office hours.2. For the love of all that is holy, come pick up your old homeworks from the box attached tomy office (RLM 11.114)3. Remember that your are required to come to office hours at least one and ask a question orexplain a solution to me.Non-homogeneous equationsWe are looking at equations of the formL[y] = y00+ p(t)y0+ q(t)y = g(t) (1)where we now drop the assumption that g(t) = 0 for all t. We have the following theorem:Theorem 1.We will often refer to c1y1(t) + c2y2(t) as the associated homogeneous solution yh(t) (the bookrefers to it as the complementary solution yc(t)) and Y (t) will be called the particular solution,sometimes denoted by yp(t). There are a few methods to solving second order non-homogeneousequations. For the moment, we will be interested in two techniques: the method of undeterminedcoefficients and variation of parameters. Both of these techniques are aimed at determining yp(t),thus they assume that one knows (or can calculate) the homogeneous solutions y1, y2. Bear inmind that there are other techniques (e.g. the Laplace transform, which we may cover later in thecourse) which compute the entire solution at once (i.e. compute yhand yptogether).1Method of Undetermined CoefficientsThe method of undetermined coefficients (UC) is predicated on the idea of guessing the formof ypbased on the form of g(t). In our guess, we have unknown constants – the undeterminedcoefficients – which we solve for by plugging our guess for ypinto our differential equation anddeducing the values of our unknowns. Here are some key ideas to bear in mind when using UC:1.2.The first step in using UC is to first determine y1, y2. With that in hand, we then think of g(t)as the s um g(t) = g1(t) + · · · + gn(t) where gi(t) are either exponential, cosine, sine, or polynomialfunctions. For each gi, we make a guess Yi(t) and set the guess for our particular solution to beyp(t) = Y1(t) + · · · + Yn(t). Keep in mind that for Yishould not be y1or y2for any i. If our tableleads to that guess for Yi, we are in an exceptional case which is discussed below. The following istable describing the guesses one should make:For the exceptional cases , we merely use the table above to guess Yi(t) and if this is either y1ory2, we multiply Yiby t. If tYiis not y1, y2, we are done; otherwise, multiply once more to get t2Yi,which will most assuredly not be y1or y2. To finish finding y, we merely set y = c1y1+ c2y2+ ypand solve for c1and c2using the initial conditions.2Let’s see some examples:y00+ 3y0+ 2y = 3e−4t, y(0) = 1, y0(0) = 0y00+ 2y0+ 2y = 2 cos 2t, y(0) = −2, y(0) = 0Method of Variation of ParametersVariation of paramete rs (VP) is a more general method than UC, but requires that certainintegrals exist and are computable. We begin as before with the differential e quation L[y] =y00+ p(t)y0+ q(t)y = g(t). As in UC, we need to be able to determine the homogenous solutionsy1, y2so p, q will either be constants or y1, y2will be given. The idea behind VP is to guess thatthe particular solution is yp(t) = u1(t)y1(t) + u2(t)y2(t) where u1, u2are unknown functions of t.Plugging this into the differential equation, we are ultimately led to conclude (by details madeexplicit in the text) that the particular solution is3Let’s see some examples:y00+ 2y0+ y = t5ety00+ y =