odinary differntial equations

Question

solve   (D^2-6D+9)y=7e^-2x

Answer 1

Same as y"-6y'+9y=7e^-2x.

First solve y"-6y'+9y=0 to find the characteristic solution y_c.

Use r²-6r+9=0=(r-3)²⇒y_c=Ae^3x+Bxe^3x.

CHECK: y_c'=3Ae^3x+3Bxe^3x+Be^3x; y_c"=9Ae^3x+9Bxe^3x+3Be^3x+3Be^3x=9Ae^3x+9Bxe^3x+6Be^3x.

y_c"-6y_c'+9y_c=9Ae^3x+9Bxe^3x+6Be^3x-6(3Ae^3x+3Bxe^3x+Be^3x)+9(Ae^3x+Bxe^3x)=0, because all terms cancel.

The particular solution has to involve e^-2x so let y_p=ae^-2x, then y_p'=-2ae^-2x and y_p"=4ae^-2x.

y_p"-6y_p'+9y_p=4ae^-2x+12ae^-2x+9ae^-2x=7e^-2x, so 25a=7, a=7/25, y_p=7e^-2x/25.

The solution is y=y_c+y_p=Ae^3x+Bxe^3x+7e^-2x/25. (A and B are arbitrary constants.)