Answer to Question #302578 in Differential Equations for haru

Question #302578

Find the general solution of the following differential equations using method of undetermined coefficients: (iii) y''− 2y'+y = xsinx.


1
Expert's answer
2022-02-28T16:11:22-0500

Corresponding homogeneous differential equation


y2y+y=0y''−2y'+y =0

Characteristic (auxiliary) equation


r22r+1=0r^2-2r+1=0

r1=r2=1r_1=r_2=1

The general solution of the homogeneous differential equation is


yh=c1ex+c2xexy_h=c_1e^x+c_2xe^x

Find the particular solution of the non homogeneous differential equation


yp=Axsinx+Bxcosx+Csinx+Dcosxy_p=Ax\sin x+Bx\cos x+C\sin x+D\cos x

yp=Asinx+Axcosx+BcosxBxsinxy_p'=A\sin x+Ax\cos x+B\cos x-Bx\sin x

+CcosxDsinx+C\cos x-D\sin x

yp=2AcosxAxsinx2Bsinxy_p''=2A\cos x-Ax\sin x-2B\sin x

BxcosxCsinxDcosx-Bx\cos x-C\sin x-D\cos x

Substitute


2AcosxAxsinx2Bsinx2A\cos x-Ax\sin x-2B\sin x

BxcosxCsinxDcosx-Bx\cos x-C\sin x-D\cos x

2Asinx2Axcosx2Bcosx+2Bxsinx-2A\sin x-2Ax\cos x-2B\cos x+2Bx\sin x

2Ccosx+2Dsinx+Axsinx+Bxcosx-2C\cos x+2D\sin x+Ax\sin x+Bx\cos x

+Csinx+Dcosx=xsinx+C\sin x+D\cos x=x\sin x

2B=12B=1

2A=0-2A=0

2BC2A+2D+C=0-2B-C-2A+2D+C=0

2AD2B2C+D=02A-D-2B-2C+D=0


A=0,B=1/2,C=1/2,D=1/2A=0, B=1/2, C=-1/2, D=1/2

yp=x2cosx12sinx+12cosxy_p=\dfrac{x}{2}\cos x-\dfrac{1}{2}\sin x+\dfrac{1}{2}\cos x

The general solution of the non homogeneous differential equation is


y=c1ex+c2xex+x2cosx12sinx+12cosxy=c_1e^x+c_2xe^x+\dfrac{x}{2}\cos x-\dfrac{1}{2}\sin x+\dfrac{1}{2}\cos x




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