= Solution
The <Neumann boundary conditions> select the spatial <eigenfunctions> $\cos nx$, $n=0,1,\ldots$, of the separated <wave equation>. Indeed separation gives $X''+\lambda X=0$ with $X'(0)=X'(\pi)=0$. Integration by parts gives $\lambda\int X^2=\int X'^2\ge0$. The zero value yields the constant mode; for positive $\lambda$, the first boundary condition removes the sine term and the second requires $\sqrt\lambda=n\in\mathbb N$. For $n>0$, the temporal factor is $\cos nct$: the sine factor vanishes because the initial velocity is zero. The zero mode has constant displacement, with no term linear in time for the same reason.
The constant <Fourier cosine series> coefficient is the mean $\pi^{-1}\int_0^\pi bx\,dx=b\pi/2$. The remaining coefficients are
$$
a_n=\frac{2b}{\pi}\int_0^\pi x\cos nx\,dx
=\frac{2b}{\pi n^2}\bigl((-1)^n-1\bigr).
$$
Only odd modes survive. Thus \b[the displacement is]
$$
\boxed{y(x,t)=\frac{b\pi}{2}-\frac{4b}{\pi}\sum_{k=0}^{\infty}
\frac{\cos((2k+1)x)\cos((2k+1)ct)}{(2k+1)^2}.}
$$
The series for displacement converges absolutely and uniformly. The initial velocity is zero in the finite-energy sense, and the initial displacement is $bx$. There is an endpoint compatibility qualification: when $b\ne0$, the initial slope does not satisfy the <Neumann boundary conditions> at $t=0$. The answer is therefore a finite-energy <weak solution>, piecewise classical between the propagating corners, rather than a globally twice continuously differentiable solution up to the initial endpoints. The mode expansion implements reflection at those endpoints.
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