u=F(xy) where u is harmonic

[quiet lagoon]

how do you call these sort of questions ?
given u=F(xy) find all u functions that are harmonic
it could be anything else in place of (xy), this is just an example
more generally u = F(x,y)

I know how to solve it but I can't find example problems and I dont know how to search for them, I have only seen it in class so far.

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[open latch]

I dont get the question tbh

Like a function $u:=F(x,y)$ is harmonic if $u_{xx}+u_{yy}=0$

[boreal peak]

Omegabet_

[quiet lagoon]

I want to search for exercises of this type
u=F(x,y) for example u=F(x+y)
And I must find the functions u such that they are harmonic

Mb i should have written u=F(g(x,y)) instead of u=F(x,y)

[open latch]

right, so for instance $u=F(x+y)\to u_{xx}+u_{yy}=F''(x+y)+F''(x+y)\equiv 0$, so you require $F''(x+y)=0$, hence $F(t)=at+b$

[boreal peak]

Omegabet_

[quiet lagoon]

Exactly

[open latch]

so the only harmonic functions of the form F(x+y) are a(x+y)+b (ie specific planes)

[quiet lagoon]

Yes

But idk where to find such exercises

[open latch]

any PDE book probably?

but you just pick a g and see what happens

[quiet lagoon]

I was thinking maybe there was a name for such problems

And yes I could just make my own functions but I was looking for a sort of course that discussed it

[open latch]

"determine harmonic functions of the form X"

like problems dont really have names perse

[quiet lagoon]

Yes, that should suffice thank you

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