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Chapter 10: Problem 12

$$\begin{array} { l } { y ^ { \prime \prime } + \lambda y = 0 ; \quad 0 < x < 2 \pi } \\ { y ( 0 ) = y ( 2 \pi ) , \quad y ^ { \prime } ( 0 ) = y ^ { \prime } ( 2 \pi ) } \end{array}$$

Short Answer

Expert verified
The eigenvalues are \(\lambda_m = -m^2\) (for \(m\) as an integer) and the corresponding eigenfunctions are \(y_m(x) = B \sin(mx)\).

Step by step solution

01

Solve the differential equation

Begin by solving the homogeneous ODE \(y'' + \lambda y = 0\). For \(\lambda = 0\), the solution is \(y(x) = Ax + B\).For \(\lambda < 0\), let \(\lambda = - \mu^2\), where \(\mu > 0\). The solution is \(y(x) = A \cos(\mu x) + B \sin(\mu x)\).For \(\lambda > 0\), let \(\lambda = k^2\), where \(k > 0\). The solution is \(y(x) = A e^{kx} + B e^{-kx}\).
02

Apply the boundary conditions

Next step is to apply the periodic boundary conditions \(y(0)=y(2\pi)\) and \(y'(0)=y'(2\pi)\) to find the corresponding eigenvalues \(\lambda\) and eigenfunctions \(y(x)\). For \(\lambda = 0\), there is no nontrivial solution that meets the boundary conditions.For \(\lambda < 0\), substituting \(\mu = n\) into the boundary conditions, we get \(A=0\) and \(n= m\), where \(m\) is an integer. Thus, the corresponding eigenvalues and eigenfunctions are \(\lambda_m = -m^2\) and \(y_m(x) = B \sin(mx)\).For \(\lambda > 0\), there are no real solutions that meet the boundary conditions.
03

Conclusion

No nontrivial solutions exist for \(\lambda = 0\) or \(\lambda > 0\). The only solutions are for \(\lambda < 0\), which are \(\lambda_m = -m^2\) and the corresponding eigenfunctions \(y_m(x) = B \sin(mx)\), where \(m\) is an integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Homogeneous ODE
A homogeneous ordinary differential equation, or simply homogeneous ODE, is a type of differential equation in which the function and its derivatives are equated to zero. In mathematical terms, for a second-order ODE, it can be represented as \( y'' + \lambda y = 0 \). Homogeneous means that each term in the equation involves the function or its derivatives in a linear form, and there are no added constants or non-zero right-hand sides.
  • Homogeneous ODEs have the characteristic feature of having solutions that can be added together to form new solutions.
  • These equations also have a direct relation with eigenvalues and eigenfunctions, which are fundamental in solving boundary value problems.
  • The general approach to solving these includes finding two linearly independent solutions and forming a general solution as their linear combination.
To solve a homogeneous ODE, one must consider different cases for \( \lambda \). Each case will guide us to different types of solutions depending on whether \( \lambda \) is zero, positive, or negative. For example:- For \( \lambda = 0 \), solutions look like \( y(x) = Ax + B \), linear in nature.- For \( \lambda > 0 \), solutions involve exponential functions.- For \( \lambda < 0 \), solutions involve trigonometric functions such as sine and cosine.
Periodic Boundary Conditions
Periodic boundary conditions are a specific type of boundary condition used to solve differential equations. These conditions require the solution to be continuous and to repeat itself after a certain period. In the exercise, periodic boundary conditions are given by \( y(0) = y(2\pi) \) and \( y'(0) = y'(2\pi) \). Under these conditions, the function and its derivative must have the same values at the beginning and the end of the interval.
  • This concept is most applicable to physical problems where a function should be periodic, such as structuring solutions to vibrational problems or wave equations.
  • By applying these conditions, we seek solutions that naturally fit the oscillatory nature of many physical systems.
  • This repetition means any potential solution must match or mirror itself over the specified interval \([0, 2\pi]\).
The imposition of periodic boundary conditions filters out non-fitting solutions when studying eigenvalue problems associated with differential equations. Solutions are then based not only on solving the ODE but ensuring that they satisfy these periodic necessities.
Eigenvalues and Eigenfunctions
Eigenvalues and eigenfunctions are fundamental concepts when dealing with differential equations, particularly in the solutions to boundary value problems like the one presented.
  • Eigenvalues \( \lambda \) are special values that permit non-trivial solutions to differential equations subject to accompanying boundary conditions.
  • Eigenfunctions are the corresponding solutions \( y(x) \) that satisfy the ODE with these eigenvalues.
  • For differential equations with periodic boundary conditions, eigenfunctions help describe solutions that naturally fit within the repeating or oscillatory frame of reference.
Finding eigenvalues and eigenfunctions involves analyzing the conditions under which solutions to the ODE exist. - These special solutions form a basis that provides insight into the behavior of more complex systems, allowing for their expansion or representation.In our specific scenario, only \( \lambda < 0 \) gave valid eigenvalues \( \lambda_m = -m^2 \) with trigonometric eigenfunctions \( y_m(x) = B \sin(mx) \), where \( m \) is an integer. This highlights that within the framework of periodic boundary conditions, only sine functions satisfy the repetitive criteria set by the boundaries.

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Most popular questions from this chapter

Verify d' Alembert's solution \((32)\) to the initial value problem \((26)-(28)\) when \(f(x)\) has a continuous second derivative and \(g(x)\) has a continuous first derivative by substituting it directly into the equations. $$ \frac{\partial^{2} u}{\partial t^{2}} $$ \(=\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}\), \(-\infty<\)x\(<\infty\), \(t>0\), $$ u(x, 0)=f(x) $$ \(-\infty<\)x\(<\infty\), $$ \frac{\partial u}{\partial t}(x, 0) $$, \(=g(x)\), \(-\infty<\)x\(<\infty\) for the given functions \(f(x)\) and \(g(x)\)

$$ \frac{\partial^{2} u}{\partial t^{2}} $$$=\frac{\partial^{2} u}{\partial x^{2}}+t x\(, \)0\(<\)x\(<\)\pi\(, \)t>0\( $$u(0, t)=u(\pi, t)=0, \quad t>0$$ $$ u(x, 0) $$$=\sin x\), \(0\)<\(x\)<\(\pi\), $$ \frac{\partial u}{\partial t}(x, 0) $$ \(=5 \sin 2 x-3 \sin 5 x\), \(0\)<\(x\)<\(\pi\)

\(f(x)=\left\\{\begin{array}{ll}{0,} & {0

Vibrating Drum. A vibrating circular membrane of unit radius whose edges are held fixed in a plane and whose displacement \(u(r, t)\) depends only on the radial distance \(r\) from the center and on the time \(t\) is governed by the initial boundary value problem. $$ \frac{\partial^{2} u}{\partial t^{2}} $$ \(\alpha^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)\), \(0\)<\(r\)<1,\( \quad t>0\), $$ u(1, t)=0, \quad t>0 $$ $$ u(r, t) \quad \text { remains finite as } r \rightarrow 0^{+} $$ Show that there is a family of solutions of the form $$ u_{n}(r, t)=\left[a_{n} \cos \left(k_{\pi} \alpha t\right)+b_{n} \sin \left(k_{n} \alpha t\right)\right] J_{0}\left(k_{\pi} r\right) $$ where \(J_{0}\) is the Bessel function of the first kind of order zero (see page 478\()\) and \(0\)

\(f(x)=\pi-x, \quad 0
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