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when solving PDE's of the form [tex]au_{xx} + 2bu_{xy} + cu_{yy} = 0[/tex] where [tex]ac - b^2 = 0[/tex] (i.e., parabolic)

is the solution always of the form:

[tex]u = xf_1 (\phi) + f_2(\phi) [/tex]

where

[tex] \phi[/tex] is the solution to the characteristic equation [tex] a(y')^2 -2by' + c = 0[/tex]

If not, is there a general form in this sense? (Related to the heat equation in the same way that d'Alembert's form relates to the wave equation)

Thanks, any help at all please is welcome.

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# Method of characteristics

Can you offer guidance or do you also need help?

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