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Taking the inner product with $f_m$ and using Fourier orthogonality isolates $a_m$. Since $x\sin (mx)$ is even, integration by parts givesConsequently the Fourier sine series isThe series converges to $x$ at each interior point and in the L2 norm. Its periodic extension has jumps at $\pm \pi$, where the Fourier series instead converges to the midpoint value zero. The overshoot near those jumps is the Gibbs phenomenon.

The second coordinate is already an eigenvector with eigenvalue $-2$. On the first and third coordinates the characteristic polynomial isThus the full eigenvalues are $3,-2,-2$. A corresponding orthonormal eigenbasis isTheir pairwise dot products vanish and each has norm one. Substitution checks $A{\mathbf{v}}_3=3{\mathbf{v}}_3$ and $A{\mathbf{v}}_{-2}=-2{\mathbf{v}}_{-2}$. The two-dimensional repeated eigenspace permits any orthonormal pair spanning the same plane; the displayed choice is convenient because it separates the second coordinate.

The characteristic polynomial of the homogeneous equation factors as $(r-1)(r-3)$, giving the complementary function $C_1{e}^x+C_2{e}^{3x}$. For a particular solution take $A\cos x+B\sin x$. Comparing cosine and sine coefficients gives $2A-4B=10$ and $4A+2B=0$, hence $A=1,B=-2$. ThereforeThe initial conditions imply $C_1+C_2=2$ and $C_1+3C_2=2$, so $C_1=2,C_2=0$. The required solution isIt has $y(0)=3$ and $y^{\prime }(0)=0$, and direct differentiation verifies $y^{\prime \prime }-4y^{\prime }+3y=10\cos x$.