Now we arrive at the most general case we treat here, the second order inhomogeneous linear differential equation for the function with constant coefficients
where and are real numbers, is a known function of , and is the function one would like to calculate. In the following, we become a bit more ‘physical’ and discuss the differential equation of the forced, damped linear harmonic oscillator, Eq. ( 2.1 ),
instead of Eq. ( 2.40 ). Since this means that both and in Eq. ( 2.40 ), we are not very general. Similar results can be obtained for the the general case.
For inhomogeneous equations the superposition principle is violated in a specail manner: we can easily show that the general solution of the differential equation can be written as the sum of a general solution of the related homogeneous equation, and a special solution to the inhomogeneous one. In other words if satisfies the inhomogeneous equation , then satisfies this same equation if satisfies the equation . This can be checked easily
The art of the exercise is thus in finding a special solution of the inhomogeneous problem.