Graduate Solve Troesch's Equation: Numerical Technique & Derivation

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Troesch's equation, u′′(x) = λ sinh(λu(x)), is being addressed through numerical techniques, with a request for its derivation and variable details. A user has provided an introduction, a solution, and a bibliography link for reference. The discussion emphasizes the need for clarity on the derivation of Troesch's equation to aid in numerical solving. The provided link leads to a general solution for Troesch's problem, which may assist in understanding the equation better. Overall, the focus remains on finding a comprehensive derivation and numerical approach for solving Troesch's equation.
wasi-uz-zaman
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Troesch's equation explanation
hi, i am going to solve Troesch's equation u′′(x) = λ sinh(λu(x))
by numerical technique, but i couldn't find the derivation of it- kindly anyone knows its derivation or details of variable used.
regards wasi
 
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Here is an introduction, a solution and a bibliography.
https://www.researchgate.net/publication/253644077_A_general_solution_for_Troesch's_problem/link/55a9969208ae815a04254f47/download
 
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I am observing an irregular, aperiodic noise pattern in the reflection signal of a high-finesse optical cavity (finesse ≈ 20,000). The cavity is normally operated using a standard Pound–Drever–Hall (PDH) locking configuration, where an EOM provides phase modulation. The signals shown in the attached figures were recorded with the modulation turned off. Under these conditions, when scanning the laser frequency across a cavity resonance, I expected to observe a simple reflection dip. Instead...
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