Discover published sets by community

Explore tens of thousands of sets crafted by our community.

Categories

Mathematics

Differential Equations

Bernoulli Differential Equations

Bessel's Differential Equations

Basic Differential Equations

Delay Differential Equations

Dynamical Systems and Phase Plane Analysis

First-Order Linear Differential Equations

Frobenius Method

Finite Difference Method for PDEs

Fourier Transform Methods for PDEs

Heat Equation

Laplace's Equation

Legendre's Differential Equations

Linear Second-Order Homogeneous Equations

Nonlinear Dynamics and Chaos

Perturbation Methods for Differential Equations

Poisson's Equation

Power Series Solutions of Differential Equations

Riccati Differential Equations

Sturm-Liouville Theory

Flashcards

0/8

Still learning

Solve the Sturm-Liouville problem $\frac{d}{dx}\left( e^x \frac{dy}{dx} \right) + \lambda y = 0$ for $y(0) = y'(1) = 0$ .

The general solution is $y(x) = A \sinh(\sqrt{\lambda}x) + B \cosh(\sqrt{\lambda}x)$ . To satisfy $y'(1)=0$ , we must have $\sqrt{\lambda} A \cosh(\sqrt{\lambda}) + B \sinh(\sqrt{\lambda}) = 0$ . These solutions are orthogonal with respect to the weight function $w(x) = 1$ over the interval $[0,1]$ .

Resolve the equation $\frac{d}{dx}\left(\sin(x)\frac{dy}{dx}\right) + \lambda \sin(x) y = 0$ with boundary conditions $y(0) = y(\pi) = 0$ .

The solution involves Legendre polynomials, with $y_n(x) = P_n(\cos(x))$ , which are orthogonal over $[0,\pi]$ with respect to weight function $w(x)=\sin(x)$ . The orthogonal functions satisfy $\int_{0}^{\pi} P_m(\cos(x)) P_n(\cos(x)) \sin(x) dx = 0$ for $m \neq n$ .

Determine the eigenfunctions for the Sturm-Liouville problem $\frac{d}{dx}\left( e^{-x}\frac{dy}{dx} \right) = \lambda y$ with $y(0) = 0$ and $y'(1) + y(1) = 0$ .

The general solution is of the form $y(x) = A e^{x/2} \sin(\sqrt{\lambda - 1/4}x) + B e^{x/2} \cos(\sqrt{\lambda - 1/4}x)$ . The eigenfunctions satisfying the boundary conditions are given by $y_n(x) = e^{x/2}\sin(\sqrt{\lambda_n - 1/4}(x-1))$ and they're orthogonal with respect to the weight function $w(x) = e^{-x}$ on $[0,1]$ .

For the boundary value problem $\frac{d}{dx}\left( \frac{1}{x} \frac{dy}{dx} \right) + \lambda y = 0$ with $y(1) = y'(e) = 0$ , determine the orthogonal functions.

The eigenfunctions are $y_n(x) = A_n \sin(n \pi \log x)$ , with $n$ being integers ensuring the boundary conditions. Two functions $y_m(x)$ and $y_n(x)$ are orthogonal over $[1,e]$ with respect to the weight function $w(x)=x$ if $m \neq n$ .

Find the general solution of $\frac{d}{dx}\left( p(x)\frac{dy}{dx} \right) + \lambda w(x) y = 0$ with $y(0) = y(\pi) = 0$ , where $p(x) = x$ and $w(x) = 1$ .

The general solution is given by a series of orthogonal functions $y_n(x) = A_n \sin(\frac{n}{\sqrt{x}} \pi x)$ . These functions are orthogonal with respect to the weight function $w(x) = 1$ over the interval $[0,\pi]$ .

Solve the Sturm-Liouville problem $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - \lambda^2)y = 0$ for $y(0)=0$ and $y(a)=0$ , where $a > 0$ .

The Bessel's equation of order $\lambda$ has solutions $y(x) = AJ_\lambda(x) + BY_\lambda(x)$ . To satisfy the boundary condition $y(0)=0$ , we set $B=0$ . The remaining solution $AJ_\lambda(x)$ must vanish at $x=a$ , leading to specific eigenvalues. Eigenfunctions are orthogonal with respect to the weight function $w(x)=x$ on $[0,a]$ .

Analyze the Sturm-Liouville problem given by $\frac{d^2y}{dx^2} - \lambda y = 0$ with $y'(0) = y'(L) = 0$ , where $L$ is a positive constant.

The general solution for this equation is $y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$ . Applying the Neumann boundary conditions ' $y'(0) = y'(L) = 0$ ' leads to $A=0$ and $\sqrt{\lambda} = \frac{n\pi}{L}$ , where $n$ is an integer. The eigenfunctions $y_n(x) = B \sin(\frac{n\pi}{L}x)$ are orthogonal with respect to the weight function $w(x)=1$ .

Considering $\frac{d}{dx}\left( x^2 \frac{dy}{dx} \right) - \lambda y = 0$ on the interval $[1,2]$ with boundary conditions $y(1) = 0$ and $(x^2 y'(x))|_{x=2} = 0$ . Find orthogonal functions.

The solution for orthogonal functions is of the form $y_n(x) = A_n x^{1/2} \sin(n \pi \log x)$ , which are orthogonal on the interval $[1,2]$ with respect to the weight function $w(x) = 1/x^2$ .

Know

Still learning

Click to flip

Know