= Solution
The <method of images> for the <Neumann boundary conditions> extends $bx$ evenly through zero and then with period $2\pi$. Define the <triangular wave>
$$
\boxed{P(s)=b|s-2k\pi|\quad\text{for }(2k-1)\pi\le s\le(2k+1)\pi,\ k\in\mathbb Z.}
$$
The formulas agree at their endpoints. It is even, $2\pi$-periodic, and agrees with $bx$ on $[0,\pi]$. Its <Fourier cosine series> has the coefficients in part (a). The cosine product identity then gives the <D'Alembert formula>
$$
\boxed{y(x,t)=\tfrac12\bigl(P(x+ct)+P(x-ct)\bigr).}
$$
It is the average of two oppositely traveling reflected waves. For $b>0$, the sketch rises from zero at even multiples of $\pi$ to $b\pi$ at odd multiples; for $b<0$ it is reflected vertically, and for $b=0$ it is zero.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-2-neumann-extension.png]
{title=Even periodic extension of the initial displacement for the Neumann wave equation}
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