Solve the second order ODE, giving a general solution: x'' + 2x' - 3x = 2e^-t

First find an auxiliary equation, for the complimentary function:m^2 + 2m - 3 = 0, (m+3)(m-1)=0m=1 or m=-3So the complimentary function is: x= Ae^t + Be^-3tFor the particular integral (PI), let x = Ue^-tthen x' = -Ue^-tand x'' = Ue^-tBy substituting these in : U(e^-t) - 2U(e^-t) -3U(e^-t) = 2e^-tDividing by e^-t, -4U=2, U=-0.5So the PI is: x= 0.5e^-tAnd, finally, the general solution is: x= Ae^t +Be^-3t +0.5e^-t