Find the general solution to the differential equation d^2x/dt^2 + 5 dx/dt + 6x = 4 e^-t

To solve a linear second order differential equation we first find the complementary function and then the particular integral. To find the complementary function we must find the roots of the auxiliary equation m²+ 5m + 6 = 0 which are m = -3 and m = -2 and which can be found by factorising. This means that the complementary function of the differential equation is x_c = A e^-3t+B e^-2t where A and B are constants.As 4 e^-t is to the right of the equals sign, the particular integral must be of the form x_p =C e^-t where C is a constant. We then differentiate the particular integral twice and substitute into the original differential equation. Differentiating gives dx_p/dt = -Ce^-t and d²x_p/dt² = Ce^-t. Substituting these into the differential equation gives 2C e^-t= 4e^-t, therefore C =2 and so x_p=2e^-t. Finally we add the complementary function to the particular integral to reach the general solution x = A e^-3t+B e^-2t+ 2e^-t.