[forward][back][up]

2.4 Inhomogeneous Equations

Now we arrive at the most general case we treat here, the second order inhomogeneous linear differential equation for the function y(x) with constant coefficients

\begin{eqnarray} y''(x) + py'(x) + qy(x) = f(x),& & %&(2.40) \\ \end{eqnarray}

where p and q are real numbers, f(x) is a known function of x, and y(x) 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),

\begin{eqnarray} \ddot{x}(t) + 2γ\dot{x}(t) + {ω}^{2}x(t)& =& {1\over m}f(x),\quad γ > 0.%&(2.41) \\ \end{eqnarray}

instead of Eq. (2.40). Since this means that both p > 0 and q > 0 in Eq. (2.40), we are not very general. Similar results can be obtained for the the general case.

2.4.1 Solution to inhomogeneous equations

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 {y}_{\text{special}}(x) satisfies the inhomogeneous equation y''(x) + py'(x) + qy(x) = f(x), then {y}_{\text{hom}}(x) + {y}_{\text{special}}(x) satisfies this same equation if {y}_{\text{hom}}(x) satisfies the equation y''(x) + py'(x) + qy(x) = 0. This can be checked easily

\begin{array}{cl} {y''}_{\text{hom}}(x) + {y''}_{\text{special}}(x) + p({y'}_{\text{hom}}(x) + {y'}_{\text{special}}(x)) + q({y}_{\text{hom}}(x) + {y}_{\text{special}}(x)) = & \\ {y''}_{\text{hom}}(x) + p{y'}_{\text{hom}}(x) + q{y}_{\text{hom}}(x) + {y''}_{\text{special}}(x) + p{y'}_{\text{special}}(x) + q + {y}_{\text{special}}(x) = 0 + f(x)&\text{(2.42)} \end{array}

The art of the exercise is thus in finding a special solution of the inhomogeneous problem.

[forward][back][up]