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Web Bonus Problems / Re: Web Bonus Problem –– Week 1
« on: January 07, 2018, 06:00:24 AM »Quote from: Jaisen Kuhle
Suppose x not equal to 0
When we solve the quadratic we arrive at imaginary values for x. I'm not sure if I ought to continue.
I agree with your first part. Now suppose x is not identically $0$, what function of $f_{yy}(y)$ multiply to $x$ will give you such a quadratic in $x$?
Also the matter seems not to be with possible complex values, but that in this case the quadratic formula gives a relation $x=F(y).$
Quote from: Victor Ivrii
Is it ever possible?No. Hence there is no common solution if $x$ is not identically zero. But if it is then upon substituting $u=u(y)=g(y)$ and so
$$u_{xx}=0=y^2 \implies y=0.$$
Thus $u$ can only be defined on the origin, and takes any constant value. (Though I doubt if derivatives are well defined then.)