Solution of Euler-Bernolli differential equation for a simple beam

I’m trying to solve the Euler-Bernolli differential equation for an homogeneous rectangular beam without load:

$$ EI{frac {partial ^{4}w}{partial x^{4}}}+mu {frac {partial ^{2}w}{partial t^{2}}}=0 $$

A way to solve it is to separate the variables: $$ w(x,t)=phi(x)Y(T)$$
and then to divide the differential equation by $phi(x)Y(T)$, obtaining:

$$ frac {partial ^{4}_x phi}{phi}+frac{mu}{EI} frac {partial ^{2}_t Y}{Y}=0. tag{1}$$

Clough and Penzien’s Dynamics of Structures textbook claims that since the first term in this equation is a function of $x$ only and the second term is
a function of $t$ only, the entire equation can be satisfied for arbitrary values of $x$ and $t$ only if each term is a constant in accordance with

$$ frac {partial ^{4}_x phi}{phi}=-frac{mu}{EI} frac {partial ^{2}_t Y}{Y}=a^4. tag{2}$$

Why can I separate the variables? i.e. why can I assume that my real system will follow the separated solution I built and neglect all the remaining solutions?
Why can I assume that the single terms in (1) are necessarily equal to a constant?

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Later, using the boundary conditions for the cantilevered beam, one finds the solution:

$$phi_n(x) = Aleft[(cosh beta _nx-cos beta _nx) + frac {cos beta _{n}L+cosh beta _{n}L}{sin beta _{n}L+sinh beta _{n}L}(sin beta _nx-sinh beta _nx)right]$$

Is the constant $A$ real or complex? Is it the same for every mode or it depends on $n$?