Corresponding homogeneous differential equation
y′′−2y′+y=0 Characteristic (auxiliary) equation
r2−2r+1=0
r1=r2=1 The general solution of the homogeneous differential equation is
yh=c1ex+c2xex Find the particular solution of the non homogeneous differential equation
yp=Axsinx+Bxcosx+Csinx+Dcosx
yp′=Asinx+Axcosx+Bcosx−Bxsinx
+Ccosx−Dsinx
yp′′=2Acosx−Axsinx−2Bsinx
−Bxcosx−Csinx−Dcosx Substitute
2Acosx−Axsinx−2Bsinx
−Bxcosx−Csinx−Dcosx
−2Asinx−2Axcosx−2Bcosx+2Bxsinx
−2Ccosx+2Dsinx+Axsinx+Bxcosx
+Csinx+Dcosx=xsinx
2B=1
−2A=0
−2B−C−2A+2D+C=0
2A−D−2B−2C+D=0
A=0,B=1/2,C=−1/2,D=1/2
yp=2xcosx−21sinx+21cosx The general solution of the non homogeneous differential equation is
y=c1ex+c2xex+2xcosx−21sinx+21cosx
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