Method of Undetermined Coefficients, Sine and Cosine

Method of Undetermined Coefficients (MUC), Sine and Cosine Consider the nonhomogeneous linear ODE with constant coefficients an y (n) + an−1 y (n−1 ) + · · · + a1 y 0 + a0 y = f (x) and suppose that f (x) = p(x)e αx sin(βx) (or f (x) = p(x)e αx cos(βx)), where p(x) is a polynomial of degree m and α and β are real numbers. Further, let k be the number of times that r = α + βi is a root to the characteristic equation of an y (n) + an−1 y (n−1) + · · · + a1 y 0 + a0 y = 0. Then, a set of associated functions is given by T = {x k e αx sin(βx), x k+1 e αx sin(βx), . . . , x k+m e αx sin(βx), x k e αx cos(βx), x k+1 e αx cos(βx), . . . , x k+m e αx cos(βx)}. Strategy for Finding a Particular Solution I Find the roots to the characteristic equation of the homogeneous version of the ODE and their multiplicities. I If r = α + βi is a root of multiplicity k, then that is the same k used in MUC. I If r = α + βi is not a root of the characteristic equation, then k = 0. I Find the general solution to the homogeneous problem, yh (x). I Let m be the degree of p(x). I Construct your set of associated functions, T = {x k e αx sin(βx), x k+1 e αx sin(βx), . . . , x k+m e αx sin(βx), x k e αx cos(βx), x k+1 e αx cos(βx), . . . , x k+m e αx cos(βx)}. Strategy for Finding a Particular Solution I Now, there exists a particular solution of the form yp (x) = A0 x k e αx sin(βx) + A1 x k+1 e αx sin(βx) + · · · + Am x k+m e αx sin(βx) +B0 x k e αx cos(βx) + B1 x k+1 e αx cos(βx) + · · · + Bm x k+m e αx cos(βx), for some constants A0 , A1 , . . . , Am , B0 , B1 , . . . , Bm . I Plug yp (x) into the ODE and solve for the coefficients. I By Principal of Superposition (Nonhomogeneous Version), the general solutions is given by y (x) = yh (x) + yp (x) Example Find a general solution to the IVP. y 00 − y 0 = −10 sin(2x), y (0) = 2, y 0 (0) = 5 Example Find a general solution to the IVP. y 00 + 3y 0 = (52x + 24) cos(2x), y (0) = 3, y 0 (0) = 1 Example Find the form of a particular solution to the ODE below using MUC. y 00 − 4y 0 + 13y = 4e −2x sin(3x) Example Find the form of a particular solution to the ODE below using MUC. y 00 + 4y 0 + 13y = 4e −2x sin(3x) Example Find the form of a particular solution to the ODE below using MUC. y 00 − 2y 0 + 2y = x 2 e x cos(x)