= Solution
For a positive <eigenvalue> write $\mu=\sqrt\lambda>0$. The <general solution> of the limiting <constant-coefficient differential equation> is $y=A\cos(\mu x)+B\sin(\mu x)$. The endpoint at zero gives $A=0$. A nonzero <eigenfunction> then requires $B\ne0$, and the endpoint at $\pi$ gives $\sin(\mu\pi)=0$. Consequently
$$
\boxed{\lambda_k=k^2,\qquad y_k(x)=C_k\sin(kx),\qquad k=1,2,\ldots,\quad C_k\ne0.}
$$
For real <eigenfunctions> the constants may be any nonzero real amplitudes. Requiring unit <L2 norm> would set $|C_k|=\sqrt{2/\pi}$, but that normalization is not required by the question. The lowest <eigenvalue> is $\lambda_1=1$ and its <eigenspace> is spanned by $\sin x$. For completeness, zero yields only an affine solution annihilated by both endpoints, and negative <eigenvalues> yield <hyperbolic functions> whose two zero endpoint values also force the zero solution.
Back to article page
