List of practice Questions

A $5$ cm long metal rod ($x$ in cm from $A$ to $B$) is governed by $\dfrac{\partial T}{\partial t}=D\,\dfrac{\partial^2 T}{\partial x^2}$ with $D=1.0~\text{cm}^2/\text{s}$ and both ends held at $0^\circ$C. The temperature field is \[ T(x,t)=\sum_{n=1,3,5,\ldots} C_n \sin\!\left(\frac{n\pi x}{5}\right)e^{-\beta n^{2} t}. \] Find $\beta$ (in s$^{-1}$, rounded to three decimals). 

A $5$ cm long metal rod ($x$ in cm from $A$ to $B$) is governed by $\dfrac{\partial T}{\partial t}=D\,\dfrac{\partial^2 T}{\partial x^2}$ with $D=1.0~\text{cm}^2/\text{s}$ and both ends held at $0^\circ$C. The temperature field is \[ T(x,t)=\sum_{n=1,3,5,\ldots} C_n \sin\!\left(\frac{n\pi x}{5}\right)e^{-\beta n^{2} t}. \] Find $\beta$ (in s$^{-1}$, rounded to three decimals).