The auxiliary equation is, m2−2m+5=0.
Solving for m, m=22±4−4⋅1⋅5=22±−16=1±2i
The complementary function is, C.F=ex(c1cos2x+c2sin2x)
Particular Integral∴P.I=D2−2D+51(x+5)=5(1+(5D2−2D))1(x+5)=51(1+(5D2−2D))−1(x+5)=51(1−(5D2−2D)+(5D2−2D)2+⋯)(x+5)=51(1−(5D2−2D))(x+5) (Neglecting higher differentials)=51((x+5)−51(D2(x+5)−2D(x+5)))=51(x+5+52)=251(5x+27)
The general solution is, y=ex(c1cos2x+c2sin2x)+255x+27
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