Sturm Liouville Form. If the interval $ ( a, b) $ is infinite or if $ q ( x) $ is not summable. We can then multiply both sides of the equation with p, and find.
P, p′, q and r are continuous on [a,b]; If λ < 1 / 4 then r1 and r2 are real and distinct, so the general solution of the differential equation in equation 13.2.2 is y = c1er1t + c2er2t. Share cite follow answered may 17, 2019 at 23:12 wang Such equations are common in both classical physics (e.g., thermal conduction) and quantum mechanics (e.g., schrödinger equation) to describe. The functions p(x), p′(x), q(x) and σ(x) are assumed to be continuous on (a, b) and p(x) >. We just multiply by e − x : P(x)y (x)+p(x)α(x)y (x)+p(x)β(x)y(x)+ λp(x)τ(x)y(x) =0. The boundary conditions (2) and (3) are called separated boundary. The boundary conditions require that Put the following equation into the form \eqref {eq:6}:
Where is a constant and is a known function called either the density or weighting function. If λ < 1 / 4 then r1 and r2 are real and distinct, so the general solution of the differential equation in equation 13.2.2 is y = c1er1t + c2er2t. Web so let us assume an equation of that form. The boundary conditions (2) and (3) are called separated boundary. The most important boundary conditions of this form are y ( a) = y ( b) and y ′ ( a) = y. Α y ( a) + β y ’ ( a ) + γ y ( b ) + δ y ’ ( b) = 0 i = 1, 2. However, we will not prove them all here. There are a number of things covered including: The solutions (with appropriate boundary conditions) of are called eigenvalues and the corresponding eigenfunctions. For the example above, x2y′′ +xy′ +2y = 0. Where is a constant and is a known function called either the density or weighting function.
