Corresponding homogeneous diffrential equaion
(D2−3D+2)y=0 Auxiliary equation
r2−3r+2=0
r1=1,r2=2 The general solution of the homogeneous differential equaion
yh=c1ex+c2e2x The particular solution of the nonhomogeneous differential equaion
y1=Ax2+Bx+C+Dsinx+Ecosx
y1′=2Ax+B+Dcosx−Esinx
y1′′=2A−Dsinx−Ecosx Substitute
2A−Dsinx−Ecosx
−6Ax−3B−3Dcosx+3Esinx
+2Ax2+2Bx+2C+2Dsinx+2Ecosx
=x2+sinx
2A=1=>A=21
−3+2B=0=>B=23
1−29+2C=0=>C=47
−3D+E=0
D+3E=1
D=101,E=103
The general solution of the nonhomogeneous differential equaion
y=c1ex+c2e2x+21x2+23x+47+101sinx+103cosx
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