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Calculus and Beyond Homework Help
Classification of Second-Order PDE with Constant Coefficients
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[QUOTE="HallsofIvy, post: 4526130, member: 637751"] Yes, that is true. But how could you know that (or what it means) if you don't know what a "characteristic" is? A characteristic is a curve upon which the partial differential equation separates into two ordinary differential equations. For example, if I associate the last formula (These are NOT equations because there is no "=". Did you mean "= 0"?) with "[itex]T^2- 4TX+ X^2[/itex] then by "completing the square" I get [itex]T^2- 4TX+ 4X^2- 3X^2= (T- 2X)^2- (\sqrt{3}X)^2= (T- 2X+\sqrt{3}X)(T- 2X- \sqrt{3}X)[/itex] So the "characteristics" are the curves [itex]t- (2-\sqrt{3})x= C[/itex] and [itex]t- (2+ \sqrt{3})x= C[/itex] for C any constant. Similarly, for the parabolic equation, we can write [itex]T^2+ 4TX+ 4X^2=N (T+ 2X)^2[/itex] and so have the single characteristic [itex]t+ 2x= C[/itex]. [/QUOTE]
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Classification of Second-Order PDE with Constant Coefficients
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