Find the general solution of the non-homogeneous differential equations: (D² − 1)y = sinx and (D² − 1)y = e^-x

The characteristic equation is obtained by replacing the differential operator $D$ with $r$:

Factor the quadratic equation:

The roots are $r_1 = 1$ and $r_2 = -1$. These are distinct real roots.

For distinct real roots $r_1$ and $r_2$, the general solution of a homogeneous linear differential equation with constant coefficients is given by:

Substitute the roots $r_1 = 1$ and $r_2 = -1$:

This is the general solution for the homogeneous part of both differential equations.

For equation (1):
The general solution of its associated homogeneous equation is $\boxed{y_h(x) = c_1 e^{x} + c_2 e^{-x}}$.

For equation (2):
The general solution of its associated homogeneous equation is $\boxed{y_h(x) = c_1 e^{x} + c_2 e^{-x}}$.