Answer to Question #283920 in Differential Equations for Torjan

Question

Solve (𝐷

2 − 3𝐷 + 2)𝑦 = 𝑥

2 + sin 𝑥 where 𝐷 =

𝑑

𝑑𝑥

Answer 1

Corresponding homogeneous equation

$(𝐷^2 − 3𝐷 + 2)𝑦 =0$

Characteristic (auxiliary) equation

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

$(r-1)(r-2)=0$

$r_1=1, r_2=2$

The general solution of the homogeneous differential equation

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

Find the partial solution of the nonhomogeneous differential equation in the form

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

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

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

Substitute

$2A-D\cos x-E\sin x-6Ax-3B$

$+3D\sin x-3E\cos x+2Ax^2+2Bx+2C$

$+2D\cos x+2E\sin x=x^2+\sin x$

$x^2: 2A=1$

$x^1: -6A+2B=0$

$x^0:2A-3B+2C=0$

$\cos x:D-3E=0$

$\sin x: E+3D=1$

$A=\dfrac{1}{2}, B=\dfrac{3}{2}, C=\dfrac{7}{4}$

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

The partial solution of the nonhomogeneous differential equation is

$y_1(x)=\dfrac{1}{2}x^2+\dfrac{3}{2}x+\dfrac{7}{4}+\dfrac{3}{10}\cos x+\dfrac{1}{10}\sin x$

The general solution of the nonhomogeneous differential equation is

$y(x)=c_1e^x+c_2e^{2x}+\dfrac{1}{2}x^2+\dfrac{3}{2}x+\dfrac{7}{4}$

$+\dfrac{3}{10}\cos x+\dfrac{1}{10}\sin x$