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

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