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Sturm-Liouville Theory Calculator

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Example
Created on 2024-06-20Asked by Benjamin Thompson (Solvelet student)
Solve the Sturm-Liouville problem for y′′+λy=0y'' + \lambda y = 0 with boundary conditions y(0)=0y(0) = 0 and y(π)=0y(\pi) = 0.

Solution

To solve the Sturm-Liouville problem y′′+λy=0y'' + \lambda y = 0 with boundary conditions y(0)=0y(0) = 0 and y(π)=0y(\pi) = 0: 1. **Solve the differential equation:** y′′+λy=0. y'' + \lambda y = 0. The general solution is: y(x)=Acos⁡(λx)+Bsin⁡(λx). y(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x). 2. **Apply the boundary conditions:** y(0)=0  ⟹  Acos⁡(0)+Bsin⁡(0)=A=0. y(0) = 0 \implies A \cos(0) + B \sin(0) = A = 0. So, y(x)=Bsin⁡(λx)y(x) = B \sin(\sqrt{\lambda} x). y(π)=0  ⟹  Bsin⁡(λπ)=0. y(\pi) = 0 \implies B \sin(\sqrt{\lambda} \pi) = 0. 3. **Find λ\lambda:** For B≠0B \neq 0, sin⁡(λπ)=0  ⟹  λπ=nπ  ⟹  λ=n  ⟹  λ=n2\sin(\sqrt{\lambda} \pi) = 0 \implies \sqrt{\lambda} \pi = n\pi \implies \sqrt{\lambda} = n \implies \lambda = n^2, where nn is an integer. 4. **Eigenfunctions and eigenvalues:** λn=n2,yn(x)=Bsin⁡(nx),n=1,2,3,… \lambda_n = n^2, \quad y_n(x) = B \sin(nx), \quad n = 1, 2, 3, \ldots 5. **Result:** The solutions to the Sturm-Liouville problem are: yn(x)=Bsin⁡(nx),λn=n2. y_n(x) = B \sin(nx), \quad \lambda_n = n^2. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Victoria Young on Solvelet
1. Find the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville problem −y′′+y=λy-y'' + y = \lambda y, subject to the boundary conditions y(0)=y(π)=0y(0) = y(\pi) = 0.2. Solve the Sturm-Liouville problem −y′′+(π2+λ)y=0 -y'' + (\pi^2 + \lambda)y = 0 with boundary conditions y(0)=y(1)=0 y(0) = y(1) = 0 .,
DefinitionThe Sturm-Liouville theory is an area of study in mathematical physics dealing with a certain type of second-order linear ordinary differential equation in the space of functions that are square integrable over a finite interval and satisfy certain homogeneous boundary conditions that depend in general on the eigenvalue parameter. Ex:The equation ( \( p(x)y' \) ) '+ q(x)y + \( \lambda r(x)y \) = 0 have eigen values λ and the corresponding eigen functions posses the specified boundary conditions.
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