Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSEach is a valid solution (the freedom of choice of form arises from the fact that uis specified at only one point (1, 1), and not along a continuum (say), as in boundarycondition (i)). All three are particular examples of the general solution, which may bewritten, for example, asu(x, y) =x 2 y +3+g(x 2 y),where g = g(x 2 y)=g(p) is an arbitrary function subject only to g(1) = 0. For thisexample, the forms of g corresponding to the particular solutions listed above are g(p) =0,g(p) =3p − 3, g(p) =1− p. ◭As mentioned above, in order to find a solution of the form u(x, y) =f(p) werequire that the original PDE contains no term in u, but only terms containingits partial derivatives. If a term in u is present, so that C(x, y) ≠ 0 in (20.9),then the procedure needs some modification, since we cannot simply divide outthe dependence on f(p) to obtain (20.10). In such cases we look instead fora solution of the form u(x, y) =h(x, y)f(p). We illustrate this method in thefollowing example.◮Find the general solution ofx ∂u∂x +2∂u − 2u =0. (20.15)∂yWe seek a solution of the form u(x, y) =h(x, y)f(p), with the consequence that∂u∂x = ∂h df(p) ∂pf(p)+h∂x dp ∂x ,∂u∂y = ∂h df(p) ∂pf(p)+h∂y dp ∂y .Substituting these expressions into the PDE (20.15) and rearranging, we obtain(x ∂h) (∂x +2∂h ∂y − 2h f(p)+ x ∂p )∂x +2∂p h df(p) =0.∂y dpThe first factor in parentheses is just the original PDE with u replaced by h. Therefore, ifh is any solution of the PDE, however simple, this term will vanish, to leave(x ∂p )∂x +2∂p h df(p) =0,∂y dpfrom which, as in the previous case, we obtainx ∂p∂x +2∂p ∂y =0.From (20.11) and (20.12) we see that u(x, y) will be constant along lines of (x, y) thatsatisfydxx = dy 2 ,which integrates to give x = c exp(y/2). Identifying the constant of integration c with p wefind p = x exp(−y/2). Thus the general solution of (20.15) isu(x, y) =h(x, y)f(x exp(− 1 y)), 2where f(p) is any arbitrary function of p and h(x, y) is any solution of (20.15).684