Observe that for \(k=0,1,\dots\)

\[\phi_k(x) = \sin(\pi k x/2) \]

form an orthonormal basis of function for \(L^2([0,2]\) with \(\phi(0)=\phi(2)=0\). Here the inner-product of two functions in \(f,g \in L^2([0,2]\) is

\[\langle f,g\rangle =\int_0^2 f(x)g(x) dx\]

Define the operator \(L\) acting on a function \(\phi(x)\) by

\[L\phi(x)=\frac12 \frac{\partial^2 \phi}{\partial^2x}(x) – 5 \phi(x)\]

To solve the equation

\[ \frac{\partial u}{\partial t}(x,t) = (L u)(x,t) \]

with

\[ u(0,t)=u(2,t)=0 \qquad\text{and}\qquad u(x,0)=F(x) \]

assume that \(u(x,t)\) takes the from

\[u(x,t)=\sum_{k=0}^\infty a_k(t) \phi_k(x)\]

Find the equations for the \(a_k\) and solve then find an expression for \(u(x,t)\).