Second-Order Linear EquationsA second-order linear differential equation has the formP (x)d2ydx+ Q(x)dydx+ R(x)y = G(x) (1)where P, Q, R, and G are continuous functions.If G(x) = 0 for all x, Equation (1) is called homogeneous linear equation, that is, theform of a sec ond-order homogeneous linear equation isP (x)d2ydx+ Q(x)dydx+ R(x)y = 0 (2)If G(x) 6= 0 for some x, Equation (1) is nonhomogeneous.A second-order nonhomogeneous linear differential equation with constant coefficients hasthe formay00+ by0+ cy = G(x) (3)where a, b and c are constants, a 6= 0, and G is a continuous function.The related homogeneous equationay00+ by0+ cy = 0 (4)is called the the complementary equation.Theorem If y1and y2are linearly independent solutions of Equation (2) (Equation (4)),then the general solution is given byyc(x) = C1y1(x) + C2y2(x)where C1and C2are arbitrary constants. Two solutions are called linearly independent ifneither y1nor y2is a constant multiple of the other.Theorem The general solution of the nonhomogeneous differential equation (1) (Equation(3)) can be written asy(x) = yc(x) + yp(x)where yc(x) is the general solution of the complementary Equation (2) (Equation (4)) andyp(x) is a particular solution of Equation (1) (Equation (3)).An initial-value problem for a second-order equation consists of finding a solution y of thedifferential equation that also satisfies initial conditions of the form: y(x0) = y0, y0(x0) = y1,where y0and y1are given constants.A b oundary-value problem for a second-order equation consists of finding a solution y ofthe differential equation that also satisfies boundary conditions of the form: y(x0) = y0, y(x1) = y1,where y0and y1are given constants.1Solutions of ay00+ by0+ cy = 0Roots of ar2+ br + c = 0 General solutionr1, r2real and distinct yc(x) = C1er1x+ C2er2xr1= r2= r yc(x) = C1erx+ C2xerxr1, r2complex: α ± iβ yc(x) = eαx(C1cos βx + C2sin βx)Two methods to find a particular solution yp(x):• The Method of Undetermined Coefficients.• The Method of Variation of Parameters.Undetermined CoefficientsG(x) = First try yp(x) =Cnxn+ · · · + C1x + C0Anxn+ · · · + A1x + A0CekxAekxC cos kx + D sin kx A cos kx + B sin kxModification: If any term of yp(x) is a solution of the complimentary equation,multiply ypby x (or by x2if necessary).Variation of ParametersWe look for a particular solution of the nonhomogeneous equation ay00+ by0+ cy = G(x) ofthe formyp(x) = u1(x)y1(x) + u2(x)y2(x)where y1(x) and y2(x) are two linearly independent solutions of the complementary equationay00+ by0+ cy = 0. The functions u1(x) and u2(x) are solutions of the following system ofequations:(u01y1+ u02y2= 0a (u01y01+ u02y02) = G(x)The method is called is variation of parameters because we have varied the parameters C1and C2to make them