I don't understand why the coefficients of the sin and cos terms in G(t) (in the red box in picture 1) depends on "n" Why don't we simply choose them to be equal to 1?
kreyszig's advanced engineering mathematics (10th edition) "Sec 12"
2026-03-26 04:53:45.1774500825
Why the coefficients of G(t) depends on "n" in solving PDE wave equation by separation of variables
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Because ultimately you want to compose these separable solutions into an sum $$ u(x,t)=\sum_n u_n(x,t)=\sum_n \Bigl[B_nu_{n,c}(x,t)+B_n^*u_{n,s}(x,t)\Bigr] $$ that hopefully can be proven to contain the solution for some specification of the parameters $B_n, B_n^*$. These coefficients are usually fixed by the initial and boundary conditions.