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- What does the notation $xf(x)$ and $f_{xxx}$ or $f_{xxy}$ mean?
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- Limit of $\\frac{x^c-c^x}{x^x-c^c}$ as $x \\rightarrow c$
My question is: Show that $\lim_{x \rightarrow c} \frac{x^c-c^x}{x^x-c^c}$ exists and find its value Because the limit is 0 0 I've tried using L'Hopital's rule, but every time I differentiate it I
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- multivariable calculus - Solving $u_{t}+u_{xxx} = u\sin{x}$ using 1 . . .
This is a very partial answer Maybe you should solve the problem exactly as Airy's equation is solved Using that $$ u\sin x = \frac u{2i}(e^{ix}-e^{-ix}), $$ take Fourier transform to the equation to obtain $$ \partial_t \hat u(t,\xi) +(i\xi)^3 \hat u(t,\xi) =\frac 1{2i}(\hat u(t,\xi-1)-\hat u(t,\xi+1)), $$ which simplifies to $$ \partial_t \hat u(t,\xi) = i\xi^3 \hat u(t,\xi) + \frac 1{2i
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