Question

As discussed in the class, if the set $\phi_n(x)$_{n=1}^{\infty}\) is orthonormal with respect to a positive weight function $w(x)$ on $[a, b]$, that is, $\int_a^b \phi_n(x)\phi_m(x)w(x)dx = \begin{cases} 0, & m \neq n, \\ 1, & m = n, \end{cases}$ then with any piecewise continuous function $f$, we can identify an orthogonal $\cdot$ expansion $f(x) \sim \sum_{n=1}^{\infty} c_n\phi_n(x),$ where $c_n = \int_a^b f(x)\phi_n(x)w(x)dx.$ The eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to $\sigma(x)$ for a regular Sturm-Liouville boundary value problem. Hence, we can use the eigenfunctions to form an orthonormal system with respect to $\sigma(x)$ simply by normalizing (scaling) each eigenfunction $\phi_n(x)$ so that $\int_a^b \phi_n^2(x)\sigma(x)dx = 1.$ a) Construct an orthonormal system of eigenfunctions corresponding to the regular Sturm-Liouville boundary value problem $\begin{cases} y'' + \lambda y = 0, \\ y(0) = 0, \\ y'(\pi) = 0. \end{cases}$ b Express $f(x) = \begin{cases} 2x/\pi, & 0 \le x \le \pi/2 \\ 1, & \pi/2 \le x \le \pi, \end{cases}$ in an eigenfunction expansion using the orthonormal system of eigen- functions obtained in part a).

          As discussed in the class, if the set \(\phi_n(x)\)_{n=1}^{\infty}\) is orthonormal
with respect to a positive weight function \(w(x)\) on \([a, b]\), that is,
\(\int_a^b \phi_n(x)\phi_m(x)w(x)dx = \begin{cases} 0, & m \neq n, \\ 1, & m = n, \end{cases}\)
then with any piecewise continuous function \(f\), we can identify an orthogonal
$\cdot$ expansion
\(f(x) \sim \sum_{n=1}^{\infty} c_n\phi_n(x),\)
where
\(c_n = \int_a^b f(x)\phi_n(x)w(x)dx.\)
The eigenfunctions corresponding to distinct eigenvalues are orthogonal with
respect to \(\sigma(x)\) for a regular Sturm-Liouville boundary value problem. Hence,
we can use the eigenfunctions to form an orthonormal system with respect
to \(\sigma(x)\) simply by normalizing (scaling) each eigenfunction \(\phi_n(x)\) so that
\(\int_a^b \phi_n^2(x)\sigma(x)dx = 1.\)
a) Construct an orthonormal system of eigenfunctions corresponding to
the regular Sturm-Liouville boundary value problem
\(\begin{cases} y'' + \lambda y = 0, \\ y(0) = 0, \\ y'(\pi) = 0. \end{cases}\)
b Express
\(f(x) = \begin{cases} 2x/\pi, & 0 \le x \le \pi/2 \\ 1, & \pi/2 \le x \le \pi, \end{cases}\)
in an eigenfunction expansion using the orthonormal system of eigen-
functions obtained in part a).

As discussed in the class, if the set (x)n=1^∞ is orthonormal
with respect to a positive weight function w(x) on [a, b], that is,
^b (x)(x)w(x)dx = 0, m ≠ n,
1, m = n,
then with any piecewise continuous function f, we can identify an orthogonal
· expansion
f(x) ∼∑n=1^∞ cn(x),
where
cn = ^b f(x)(x)w(x)dx.
The eigenfunctions corresponding to distinct eigenvalues are orthogonal with
respect to σ(x) for a regular Sturm-Liouville boundary value problem. Hence,
we can use the eigenfunctions to form an orthonormal system with respect
to σ(x) simply by normalizing (scaling) each eigenfunction (x) so that
^b ^2(x)σ(x)dx = 1.
a) Construct an orthonormal system of eigenfunctions corresponding to
the regular Sturm-Liouville boundary value problem
y” + λ y = 0,
y(0) = 0,
y'(π) = 0.
b Express
f(x) = 2x/π, 0 ≤ x ≤π/2
1, π/2 ≤ x ≤π,
in an eigenfunction expansion using the orthonormal system of eigen-
functions obtained in part a).

Added by Francisco B.

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