DO4

Find the general solution of $(D^2-1)y = 2x + e^{2x}\,$ .

The auxiliary equation is $m^2-1=0\,$ with roots $m=-1,1\,$ .

The complimentary solution is $y_c = Ae^x + Be^{-x}\,$ .

Guess the particular solution based on the inhomogenous term of the DE: $y_p = c_1 x + c_2 e^{2x}\,$ .

$(D^2-1)y_p = 4c_2e^{2x} - c_1 x - c_2 e^{2x} = 2x + e^{2x}\,$ .

Equate coefficients.

$3c_2 = 1,\,\,\, -c_1 = 2\,$

So $c_2 = 1/3,\,\,\, c_1 = -2\,$ .

$y_p(x) = -2x + \frac{1}{3}e^{2x}\,$

The general solution is $y = y_c + y_p\,$ .

$y(x) = Ae^x + Be^{-x} - 2x + \frac{1}{3}e^{2x}\,$