Particular Solutions for PDEs

SO that a repeated root A = 1 occurs. The general solution is 

Although we have derived general solutions, nothing has been said about the specific functional form taken by / and g, since these are completely arbitrary, other than the requirement that they must be continuous, with continuous derivatives. For example, considering the parabolic equation given in (10.59), the specific function fix + y) = c sinh(x + y) is seen, by substitution, to satisfy the equation. The discussion in this section has determined only the arguments, not specific functions. 

the remainder of this chapter deals with solving linear, homogeneous equations, using the following three approaches to reduce PDEs to ODEs. 

Combination of Variables Sometimes called a similarity transform, this technique seeks to combine all independent variables into one variable, which may then produce a single ODE. It is applicable only to cases where 

Separation of Variables This is the most widely used method in applied mathematics, and its strategy is to break the dependent variable into component parts, each depending (usually) on a single independent variable invariably, it leads to a multiple of particular solutions. 

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