Exercise 8.5.1

Question

\begin{enumerate}[label=(\alph*)] 
\item  Verify that
$$u(x,t) = b_n \sin(nx) \cos(nt)$$
satisfies equations (1), (2), and (3) for any choice of $n \in \mathbf{N}$ and $b_n \in \mathbf{R}$. What goes wrong if $n \notin \mathbf{N}$?
\item Explain why any finite sum of functions of the form given in part (a) would also satisfy (1), (2), and (3).
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item
$$\frac{\partial^2 u}{\partial x^2} = -b_n n^2 \sin(nx) \cos(nt) = \frac{\partial^2 u}{\partial t^2}$$
$$u(0, t) = b_n \sin(0) \cos (nt) = 0$$
$$u(\pi, t) = b_n \sin(n \pi) \cos (nt) = 0$$
(Note that the above equation is no longer true if $n \notin \mathbf{N}$.)
$$\frac{\partial u}{\partial t}(x,0) = -b_n \sin(nx) \sin (0) = 0$$
\item The differential equation itself and the boundary conditions are linear; that is, if $u_1$ and $u_2$ both satisfy equations (1) through (3), then so does $c_1 u_1 + c_2 u_2$ for any $c_1, c_2 \in \mathbf{R}$.
 \end{enumerate}

Exercise 8.5.1

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Exercise 8.5.1

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