A string of length $L$ fixed at $x=0$ and $x=L$ and released at time time $t=0$, the transverse displacement at a position $x$ along the string is given by: $y(x,0)=Ax(L-x)$
Assuming that the string is released from rest, write down an expression for $y(x, t)$, the transverse displacement profile of the string at an arbitrary later time.
My solution:
The initial displacement $y(x, 0)$ is can be represented as a sum of harmonic $y(x,0)=∑_{(n=1)}^∞a_n y_n (x) $ where $y_n(x)=\sin(nπx/L)$ and $a_n=\frac{4AL^2 (1-(-1)^n )}{(π^3 n^3 )}$.
This is the part i am struggling with:
$y(x,t)=∑_{(n=1)}^∞(\frac{4AL^2 (1-(-1)^n )}{(π^3 n^3 )})(\sin(nπx/L)) \cos(πvnt/L)$