# Tagged Questions

The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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### Finding a possible region where this PDE has a solution

Consider the problem $$xu_t+u_x = 0, \quad u(0,x) = \sin x.$$ We're asked to prove that the problem doesn't have a solution defined in all of $\Bbb R^2$, and to give a possible open set in $\Bbb R^2$ ...
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### Find the equation of the common part of two objects

How to find the equation of the intersection curve of the ball $x^2 + y^2 + z^2 = 4a^2$ (1)and the cylinder $x^2+y^2=2ax(a>0)$(2)? let (1)-(2), we can get $$z^2+2ax-4a^2=0$$ but this is not ...
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### Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
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### Is the function from the Cauchy functional equation, $f(x+y)=f(x)+f(y)$ injective?

It's obviously not injective in the case of $f(x)=0$. I'm wondering if it's injective in all other cases. The other linear solutions of the form $f(x)=c\cdot x$ where $c$ is some constant are ...
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### Sufficiency of the condition $f(x) = f(x^3)$ for $f$ to be even or constant
I've been playing around with some aspects of basic functions, and I reached a function that seemed a bit peculiar. Consider $\forall x \in \mathbb{R}$ a function $f:\mathbb{R} \rightarrow \mathbb{R}$ ...
### Solving functional equation $f(4x)-f(3x)=2x$
Given that $f(4x)-f(3x)=2x$ and that $f:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function, find $f(x)$. My thoughts so far: subtituting $\frac{3}{4}x$, $\left(\frac{3}{4}\right)^2x$, \$\left(\...