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Nonhomogeneous Second-Order Differential Equations

Nonhomogeneous Second-Order Differential EquationsTo solveay +by +cy=f(x) we first consider the solution of the formy=yc+ypwhereycsolves the differential equaitonay +by +cy= 0 andypsolves the differential equationay +by +cy=f(x).Since the derivative of the sum equals the sum of the derivatives, we will have a finalsolution of 0 +f(x) which givesf(x).Therefore, we follow a pattern forypto yieldf(x) when each ofy,y andy are substi-tuted into the above Case I: Iff(x) is of the formf(x) =P(x)ekxthen we set:yp=Q(x)ekxwhereQ(x) is the general polynomial of the same degree asP(x).For example if the differential equation is set equal to:(a)f(x) =x2e3x. Setyp= (Ax2+Bx+C)e3x.(b)f(x) =x3 4x+ 1. Setyp=Ax3+Bx2+Cx+D.(c)f(x) = 6e 2x. Setyp=Ae we substituteyp,y p, andy pand solve for the Case II: Iff(x) is of the formf(x) =P(x)ekxsin (mx) orf(x) =P(x)ekxcos (mx),then we set:yp=Q(x)ekxcos (mx) +R(x)ekxsin (mx) whereQ(x) andR(x) are both generalpolynomials of the same degree asP(x).

Nonhomogeneous Second-Order Differential Equations To solve ay′′ +by′ +cy = f(x) we first consider the solution of the form y = y c +yp where yc solves the differential equaiton ay′′ +by′ +cy = 0 and yp solves the differential equation ay′′ +by′ +cy = f(x). Since the derivative of the sum equals the sum of the derivatives, we will have a final

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