Particular Integrals of Linear Differential Equations

Extract

In elementary courses on differential equations the standard method of obtaining particular integrals is probably that of the variation of parameters or, as it is sometimes rather paradoxically called, the variation of constants. In more sophisticated courses the idea of the Green’s function is introduced. This has many advantages, not the least of which is that the method for finding particular integrals based on such a function is much better motivated. Moreover, it is possible to fit in given boundary conditions, homogeneous or otherwise, as part of the process and this avoids artificially breaking the problem up into two parts, in the second of which a plethora of arbitrary constants is related to given boundary or initial conditions. A disadvantage of Green’s function methods is that, if they are to be properly exploited, it is necessary to introduce the concept of the delta function.