Answer to Question #215963 in Differential Equations for killa

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

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