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ad2ydx2+bdydx+cy=F(x)a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = F(x)adx2d2y+bdxdy+cy=F(x)
Find the complementary function (C.F.)
Assume y=emxy = e^{mx}y=emx leading to m2−5m+6=0m^2 - 5m + 6 = 0m2−5m+6=0
Roots are m=2m = 2m=2 and m=3m = 3m=3 (factors: (m−2)(m−3)=0(m-2)(m-3)=0(m−2)(m−3)=0).
y=C1er1x+C2er2xy = C_1e^{r_1x} + C_2e^{r_2x}y=C1er1x+C2er2x for roots r1,r2r_1, r_2r1,r2.
y=λsin(3x)+μcos(3x)y = \lambda\sin(3x) + \mu\cos(3x)y=λsin(3x)+μcos(3x)
Set coefficients of like terms equal to each other.
Use the same degree polynomial as the non-homogeneous term.
y=yh+ypy = y_h + y_py=yh+yp
Assume y=λxe2xy = \lambda x e^{2x}y=λxe2x if e2xe^{2x}e2x is in C.F.
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