**Orthogonal (Perpendicular) Functions**You had seen in Problem #2 of Homework #3 that the non-zero solutions $\varphi(x)$ to the ODE\[\varphi''(x) + \lambda \varphi(x) = 0, \, \text{for } 0 < x < 1,\]along also the BCs $\varphi(0) + \varphi'(0) = 0$ and $\varphi(1) + \varphi'(1) = 0$, are\[\varphi_n(x) = \begin{cases} e^{-x}, & \text{when } \lambda_0 = -1 \\ \sin(n \pi x) - n \pi \cos(n \pi x), & \text{when } \lambda_n = (n \pi)^2 \, \text{for } n = 1, 2, 3, \ldots \end{cases}\]a.) Show that\[\varphi_p \cdot \varphi_q = \int_0^1 \varphi_p(x) \varphi_q(x) dx = 0\]for $p \neq q$ and compute this dot product when $p = q$.b.) Determine the coefficients $A_n$ if the function $f(x) = 1$ is to be expanded as\[f(x) = 1 = \sum_{n=0}^{\infty} A_n \varphi_n(x) \quad \text{for all } \quad 0 < x < 1,\]where the $\varphi_n(x)$'s (for $n = 0, 1, 2, 3, \ldots$) are the functions above.

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Answered: e*, when 20 = -1 %3D P„(x) sin(nrx) –…

e*, when 20 = -1 %3D P„(x) sin(nrx) – nn cos(nax), when 2, = (nx)² for n = 1,2,3,.... a.) Show that Pp • Pq J. ,r)@,(x)dx = 0 %3D for p + q and compute this dot product when p = q.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**Orthogonal (Perpendicular) Functions**

You had seen in Problem #2 of Homework #3 that the non-zero solutions $\varphi(x)$ to the ODE

\[
\varphi''(x) + \lambda \varphi(x) = 0, \, \text{for } 0 < x < 1,
\]

along also the BCs $\varphi(0) + \varphi'(0) = 0$ and $\varphi(1) + \varphi'(1) = 0$, are

\[
\varphi_n(x) =
\begin{cases}
e^{-x}, & \text{when } \lambda_0 = -1 \\
\sin(n \pi x) - n \pi \cos(n \pi x), & \text{when } \lambda_n = (n \pi)^2 \, \text{for } n = 1, 2, 3, \ldots
\end{cases}
\]

a.) Show that

\[
\varphi_p \cdot \varphi_q = \int_0^1 \varphi_p(x) \varphi_q(x) dx = 0
\]

for $p \neq q$ and compute this dot product when $p = q$.

b.) Determine the coefficients $A_n$ if the function $f(x) = 1$ is to be expanded as

\[
f(x) = 1 = \sum_{n=0}^{\infty} A_n \varphi_n(x) \quad \text{for all } \quad 0 < x < 1,
\]

where the $\varphi_n(x)$'s (for $n = 0, 1, 2, 3, \ldots$) are the functions above.

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