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

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