Advanced Calculus fi..

Chapter 9 Ordinary Differential Equations 63514. (Method of undetermined coeficients) To obtain a solution y*(x) of the equation"(0' + l)[yl = ex,(a)we can multiply both sides by D - 1:(D- 1 ) ( + ~ l)[y] ~ = (D- l)[ex] = 0. (b)The right-hand member is 0 because ex satisfies the homogeneous equation(D - l)[v] = 0. From (b) wc obtain the characteristic equation (r - l)(r2 + 1) = 0.Accordingly, every solution of (b) has the formy = cl cosx + c2sinx + c3ex.If we substitute this expression in (a), the first two terms give 0, since they form thecomplementary function of (a). One obtains an equation for the "undetermined coefficient"cg: 2c3ex = ex, c3 = i. Accordingly, y* = ;ex is the particular solution sought.The method depends upon finding an operator that "annihilates" the right-hand memberQ(x) Such an operator can always be found if Q is a solution of a homogeneous equationwith constant coefficients: the operator L such that L[Q] = 0 is the operator sought.If Q is of form (po + plx + ... + pkxk-')eaX(~ cos bx + B sin bx), the annihilatoris L = ((D- a12 + b2)k. For a sum of such terms, one can multiply the correspondingoperators. Use the method described to find particular solutions of the followingequations:a) y" + y = sinx (annihilator is D~ + 1)b) y" + y = e2X (annihilator is D - 2)C) y" + y' - y = x2 (annihilator is D ~)d) y"+2y1+y =sin2x+eXe) y" + 4y = 25xeX I Cf .‘f) y"' - 2y' - 4y = e-X sinx15. Prove by variation of parameters or by the method of Problem 14 that a particular solutionof the equationis given byB sin (wt - a)20hX = tana= )c2 - ,2 ' O