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Calculus and Beyond Homework Help
Memorizing solutions for differential equations
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[QUOTE="Molar, post: 5552371, member: 562132"] This is interesting. I never thought it ! For what the professor gave you is a shortcut to remember. It is said in the equation [B]d[SUP]2[/SUP]y/dx[SUP]2[/SUP] = -k[SUP]2[/SUP]y[/B] itself. You have a function 'y' . Now you differentiate it two times with respect to 'x' and get back 'k' times the original function with the opposite sign. Now which function behaves like this ? Only if, $$ y = sin kθ $$ or, $$ y = cos kθ $$ or, $$ y = e^{ikx} $$ Again e[SUP]ikx[/SUP] = [I]cos kx + i sin kx.[/I] That's how you know the general solution is this. Similarly in the next equation, you get the original function back with unaltered sign. Only an exponential function (without '[I]i [/I]' in it) behaves in that way. [/QUOTE]
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Memorizing solutions for differential equations
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