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 Yearling
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Nov
25
revised Separating Partial Differential Eq
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Nov
25
comment Separating Partial Differential Eq
The separation I've shown you is correct. What is $a$ and $b$? As I told you, the relationship between $C$ and the separation constant comes from the boundary conditions. Why don't you edit your question and show us how did you derived the relation $a^2 + b^2 = C^2$? Here is a quick guide on how to typeset equations.
Nov
25
awarded  Informed
Nov
25
comment Separating Partial Differential Eq
Well, if by a relationship you mean $C = C(\lambda)$, then no. The solution of the system is \begin{align}\Theta(\theta) &= A e^{\sqrt{\lambda} x} + B e^{-\sqrt{\lambda} x} \\ R(r) &= C J_\sqrt{\lambda}(C x) + D Y_\sqrt{\lambda}(C x)\end{align} where $J_\sqrt{\lambda}$ and $Y_\sqrt{\lambda}$ are Bessel and Neumann functions of the first kind, of order $\sqrt{\lambda}$. If no boundary conditions are given, then you have an infinite collection of solutions spanned by both parameters.
Nov
25
comment Separating Partial Differential Eq
I'm guessing you're getting Helmoltz equation from separating time and space from a cylindrical wave equation. If you need to quantize $\lambda$ and $C$, then boundary conditions are needed.
Nov
25
answered Separating Partial Differential Eq
Nov
15
awarded  Yearling
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy Indeed.
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy It gets absorbed in the definition of $\hat{c}_1(k)$. I abused the notation, sorry!
Sep
10
comment Solution of a differentiation in integral form
@ComplexGuy When $t=0$, $\sin(\omega t) = 0$, and the equality follows.
Sep
10
revised Comparison between Bessel's coefficients
deleted 21 characters in body
Sep
10
answered Comparison between Bessel's coefficients
Jul
22
comment Perturbation Theory on Finite Domains
You should read about the WKB approximation. A great treatise on the asymptotics of $y'' + p(x) y = 0$ can be found in chapter 6 of Olver's Asymptotics and Special Functions.
Jul
14
comment Nonlinear equation (oscillon) comparison
@ComplexGuy en.wikipedia.org/wiki/Virtual_work
Jul
13
revised General Solution of a Differential Equation using Green's Function
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Jul
13
revised General Solution of a Differential Equation using Green's Function
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Jul
13
revised General Solution of a Differential Equation using Green's Function
added 3 characters in body
Jul
13
answered General Solution of a Differential Equation using Green's Function
Jul
12
revised Proving a triangle is a right triangle given vertices, using vector dot product
edited tags
Jul
12
comment Solve the following differential equations by converting to Clairaut's form through suitable substitutions.
@Vish.Math See my last edit but be warned that there is never a satisfactory explanation for intuitive steps. Math is experience.