Question

(D² -2D +3)y = x² -1

Answer 1

$$(D^2-2D+3)y=x^2-1$$

The homogeneous differential equation

$$(D^2-2D+3)y=0$$

The characteristic equation

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

$$r_1=1-\sqrt{2}i,r_2=1+\sqrt{2}i$$

The general solution of the homogenepus equation is

$$y_h=c_1e^x\cos(\sqrt{2}x)+c_2e^x\sin(\sqrt{2}x)$$

Find the particular solution of the nonhomogeneous differential equation

$$y_p=Ax^2+Bx+C$$

$$y_p '=2Ax+B$$

$$y_p''=2A$$

Substitute

$$2A-4Ax-2B+3Ax^2+3Bx+3C=x^2-1$$

$$3A=1=>A=\dfrac{1}{3}$$

$$-4A+3B=0=>B=\dfrac{4}{9}$$

$$2A-2B+3C=-1=>C=-\dfrac{7}{27}$$

Then

$$y_p=\dfrac{1}{3}x^2+\dfrac{4}{9}x-\dfrac{7}{27}$$

The general solution of the given nonhomogenepus equation is

$$y=c_1e^x\cos(\sqrt{2}x)+c_2e^x\sin(\sqrt{2}x)+\dfrac{1}{3}x^2+\dfrac{4}{9}x-\dfrac{7}{27}$$