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Handbook of mathematics for engineers and scienteists part 91

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 91', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 598 Linear Partial Differential Equations Example 4. Consider the heat equation dw d2w In this case we have M a-d Therefore the formal series solution has the form w x t f x k- f n x f m d f x . fe l If the function f x is taken as a polynomial of degree n the solution will also be a polynomial of degree n. For example setting f x Ax2 Bx C we obtain the particular solution w x t A x2 2at Bx C. 2 . The equation d2w n 2 M w dt2 where M is a linear differential operator just as in Item 1 has a formal solution represented by the sum of two series as t2k t2k 1 w x t 2 M k f x C 2T i TM k 9 x L k 0 v 7 k 0 v 7 where f x and g x are arbitrary infinitely differentiable functions. This solution satisfies the initial conditions w x 0 f x and dtw x 0 g x . 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions 14.3.2-1. Simplest properties of nonhomogeneous linear equations. For brevity we write a nonhomogeneous linear partial differential equation in the form L w x t 14.3.2.1 where the linear differential operator L is defined above see the beginning of Paragraph 14.3.1-1 . Below are the simplest properties of particular solutions of the nonhomogeneous equation 14.3.2.1 . 1 . If W x t is a particular solution of the nonhomogeneous equation 14.3.2.1 and w0 x t is a particular solution of the corresponding homogeneous equation 14.3.1.1 then the sum Aw0 x t w x t where A is an arbitrary constant is also a solution of the nonhomogeneous equation 14.3.2.1 . The following more general statement holds The general solution of the nonhomogeneous equation 14.3.2.1 is the sum of the general solution of the corresponding homogeneous equation 14.3.1.1 and any particular solution of the nonhomogeneous equation 14.3.2.1 . 2 . Suppose wl and w2 are solutions of nonhomogeneous linear equations with the same left-hand side and different right-hand sides i.e. L W1 i x t L W2 2 x t . Then the function w wl w2 is a solution of the equation L w i x t 2 x t . 14.3. Properties and .