Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSis of the formA(x, y) ∂u ∂u+ B(x, y) + C(x, y)u = R(x, y), (20.9)∂x ∂ywhere A(x, y), B(x, y), C(x, y) andR(x, y) are given functions. Clearly, if eitherA(x, y) orB(x, y) is zero then the PDE may be solved straightforwardly as afirst-order linear ODE (as discussed in chapter 14), the only modification beingthat the arbitrary constant of integration becomes an arbitrary function of x or yrespectively.◮Find the general solution u(x, y) ofx ∂u∂x +3u = x2 .Dividing through by x we obtain∂u∂x + 3ux = x,which is a linear equation with integrating factor (see subsection 14.2.4)(∫ ) 3expx dx =exp(3lnx) =x 3 .Multiplying through by this factor we find∂∂x (x3 u)=x 4 ,which, on integrating with respect to x, givesx 3 u = x55 + f(y),where f(y) isanarbitrary function of y. Finally, dividing through by x 3 , we obtain thesolutionu(x, y) = x25 + f(y)x . ◭ 3When the PDE contains partial derivatives with respect to both independentvariables then, of course, we cannot employ the above procedure but must seekan alternative method. Let us for the moment restrict our attention to the specialcase in which C(x, y) =R(x, y) = 0 and, following the discussion of the previoussection, look for solutions of the form u(x, y) =f(p) wherep is some, at presentunknown, combination of x and y. We then have∂u∂x = df(p) ∂pdp ∂x ,∂u∂y = df(p) ∂pdp ∂y ,682