Method of Undetermined Coefficients, Polynomials

Definition A set of associated functions, denoted by T , is a set of functions whereby a particular solution to a nonhomogeneous linear ODE is known to be a linear combination of the functions in T . Method of Undetermined Coefficients (MUC), Polynomials 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) is a polynomial of degree m. Further, let k be the number of times that r = 0 is a root to the characteristic equation of an y (n) + an−1 y (n−1) + · · · + a1 y 0 + a0 y = 0, i.e. the multiplicity of r = 0. Then, a set of associated functions is given by T = {x k , x k+1 , . . . , x k+m }. 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 = 0 is a root of multiplicity k, then that is the same k used in MUC. I If r = 0 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 f (x). I Constuct your set of associated functions, T = {x k , x k+1 , . . . , x k+m }. Strategy for Finding a Particular Solution I Now, there exists a particular solution of the form yp (x) = A0 x k + A1 x k+1 + · · · + Am x k+m , for some constants A0 , A1 , . . . , Am . 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 = −2x + 2, y (0) = 1, y 0 (0) = 2 Example Find a general solution to the IVP. y 00 + 3y 0 = 11 + 6x, y (0) = 1, y 0 (0) = 0 Example Find a general solution to the IVP. y 00 = 4 − 36x 2 + 100x 3 , y (1) = 4, y 0 (1) = 16 Example Find a general solution to the IVP. y 00 − 4y = −4x 2 + 6, y (0) = 5, y 0 (0) = 0 Example Find a general solution to the IVP. y 00 − 2y 0 + 2y = 2x 3 − 6x 2 + 6x + 2, y (0) = −1, y 0 (0) = 0