Answer to Question #283797 in Differential Equations for Pallavi

Question #283797

Solve (𝐷^2 − 3𝐷 + 2)𝑦 = 𝑥^2 + sin 𝑥 where 𝐷 =𝑑/𝑑𝑥

1
Expert's answer
2022-01-02T17:30:49-0500

Corresponding homogeneous diffrential equaion


(𝐷23𝐷+2)𝑦=0(𝐷^2 − 3𝐷 + 2)𝑦 = 0

Auxiliary equation


r23r+2=0r^2-3r+2=0

r1=1,r2=2r_1=1, r_2=2

The general solution of the homogeneous differential equaion


yh=c1ex+c2e2xy_h=c_1e^x+c_2e^{2x}

The particular solution of the nonhomogeneous differential equaion


y1=Ax2+Bx+C+Dsinx+Ecosxy_1=Ax^2+Bx+C+D\sin x+E\cos x

y1=2Ax+B+DcosxEsinxy_1'=2Ax+B+D\cos x-E\sin x

y1=2ADsinxEcosxy_1''=2A-D\sin x-E\cos x

Substitute


2ADsinxEcosx2A-D\sin x-E\cos x

6Ax3B3Dcosx+3Esinx-6Ax-3B-3D\cos x+3E\sin x

+2Ax2+2Bx+2C+2Dsinx+2Ecosx+2Ax^2+2Bx+2C+2D\sin x+2E\cos x

=x2+sinx=x^2+\sin x

2A=1=>A=122A=1=>A=\dfrac{1}{2}

3+2B=0=>B=32-3+2B=0=>B=\dfrac{3}{2}

192+2C=0=>C=741-\dfrac{9}{2}+2C=0=>C=\dfrac{7}{4}

3D+E=0-3D+E=0

D+3E=1D+3E=1

D=110,E=310D=\dfrac{1}{10}, E=\dfrac{3}{10}

The general solution of the nonhomogeneous differential equaion


y=c1ex+c2e2x+12x2+32x+74y=c_1e^x+c_2e^{2x}+\dfrac{1}{2}x^2+\dfrac{3}{2}x+\dfrac{7}{4}+110sinx+310cosx+\dfrac{1}{10}\sin x+\dfrac{3}{10}\cos x

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