Answer to Question #283797 in Differential Equations for Pallavi

Question

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

Answer 1

Corresponding homogeneous diffrential equaion

$$(𝐷^2 − 3𝐷 + 2)𝑦 = 0$$

Auxiliary equation

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

$$r_1=1, r_2=2$$

The general solution of the homogeneous differential equaion

$$y_h=c_1e^x+c_2e^{2x}$$

The particular solution of the nonhomogeneous differential equaion

$$y_1=Ax^2+Bx+C+D\sin x+E\cos x$$

$$y_1'=2Ax+B+D\cos x-E\sin x$$

$$y_1''=2A-D\sin x-E\cos x$$

Substitute

$$2A-D\sin x-E\cos x$$

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

$$+2Ax^2+2Bx+2C+2D\sin x+2E\cos x$$

$$=x^2+\sin x$$

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

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

$$1-\dfrac{9}{2}+2C=0=>C=\dfrac{7}{4}$$

$$-3D+E=0$$

$$D+3E=1$$

$$D=\dfrac{1}{10}, E=\dfrac{3}{10}$$

The general solution of the nonhomogeneous differential equaion

$$y=c_1e^x+c_2e^{2x}+\dfrac{1}{2}x^2+\dfrac{3}{2}x+\dfrac{7}{4}$$ $$+\dfrac{1}{10}\sin x+\dfrac{3}{10}\cos x$$