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

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

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