Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 31

Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).

Short Answer

Expert verified
The solvability conditions for the given boundary value problem can't be determined without additional information.

Step by step solution

01

Understanding the Problem

The given problem \(u^{\prime}(\pi / 4) = \beta\) is a boundary value problem as it specifies the value of the derivative of the function at a particular point, here at \(x=\pi / 4\). The solvability of this problem, in simplest terms, would mean a set of conditions under which a solution to the equation exists.
02

Identifying Solvability Conditions

This is a first order differential equation, and it's known an initial value problem of the form \(u^{\prime} = f(x, u)\) has a unique solution provided \(f\) and \(\partial f / \partial u\) are continuous within a region containing the initial point. However, we don't have a function f(x,u) here, the derivative at a single point \(x = pi / 4\) is given, not an interval. This limits our ability to reach a conclusion about the solvability directly.
03

Conclusion

Without additional information, no general conclusion can be given about the solvability of the given boundary problem since only the derivative value at a single point is given

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The coefficients \(C_{k}^{p}\) in the binomial expansion for \((1+x)^{p}\) are given by $$ C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k !} $$ a. Write \(C_{k}^{p}\) in terms of Gamma functions. b. For \(p=1 / 2\), use the properties of Gamma functions to write \(C_{k}^{1 / 2}\) in terms of factorials. c. Confirm your answer in part b. by deriving the Maclaurin series expansion of \((1+x)^{1 / 2}\)

Show that a Sturm-Liouville operator with periodic boundary condion \([a, b]\) is self-adjoint if and only if \(p(a)=p(b)\). [Recall that periodic dary conditions are given as \(u(a)=u(b)\) and \(\left.u^{\prime}(a)=u^{\prime}(b) .\right]\).

A solution of Bessel's equation, \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-n^{2}\right) y=0\), can be found using the guess \(y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n} .\) One obtains the recurrence relation \(a_{j}=\frac{-1}{j(2 n+j)} a_{j-2} .\) Show that for \(a_{0}=\left(n ! 2^{n}\right)^{-1}\), we get the Bessel function of the first kind of order \(n\) from the even values \(j=2 k\) : $$ J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(n+k) !}\left(\frac{x}{2}\right)^{n+2 k} $$

Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).

The coefficients \(C_{k}^{p}\) in the binomial expansion for \((1+x)^{p}\) are given by $$ C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k !} $$ a. Write \(C_{k}^{p}\) in terms of Gamma functions. b. For \(p=1 / 2\), use the properties of Gamma functions to write \(C_{k}^{1 / 2}\) in terms of factorials. c. Confirm your answer in part b. by deriving the Maclaurin series expansion of \((1+x)^{1 / 2}\)
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free