Solving a Nonlinear Differential Equation

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SUMMARY

The discussion centers on solving the nonlinear differential equation [h'(x)]^2 - h(x)·h''(x) + 1 = 0, which is proposed to describe the shape of an ideal rope modeled by the function h[x] = ch[x]. While substituting this function yields a valid solution, the user seeks methods to solve the equation without prior knowledge of the solution. Suggestions include using a series expansion method, where h(x) is expressed as a sum of terms C_n * x^n, although this approach is primarily effective for linear equations.

PREREQUISITES
  • Understanding of nonlinear differential equations
  • Familiarity with hyperbolic functions, specifically the hyperbolic cosine function (ch[x])
  • Knowledge of series expansion techniques in mathematical analysis
  • Basic proficiency in using Mathematica for solving differential equations
NEXT STEPS
  • Research methods for solving nonlinear differential equations
  • Learn about series expansion techniques and their applications in differential equations
  • Explore the capabilities of Mathematica for handling boundary conditions in differential equations
  • Study the properties and applications of hyperbolic functions in physics and engineering
USEFUL FOR

Mathematicians, physics students, and engineers interested in solving nonlinear differential equations and those seeking to understand the mathematical modeling of physical phenomena such as the behavior of ideal ropes.

Semo727
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Hello!
I tried to prove, that ideal rope (see picture in attachment) has a shape of the function ch[x]. I finished with this equation

[tex][h'(x)]^2-h(x)\cdot h''(x)+1=0[/tex]

Yes, when you try function h[x]=ch[x], you get 0 on the left side, but I have no clue how to solve this equation (find the solution without knowing the solution:rolleyes:). Even Mathematica has some problems, if I set boundary conditions. Could you please write how to solve this DE (providing it isn't too complicated, because I don't know much about solving nonlinear DE)
 

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Guess and check is the only guaranteed way of finding a solution to any DE. You might be able to use a series expansion here (guess solution of form h(x) = sum(C_n * x^n) and plug in, but I'm not sure. That method is guaranteed to work for linear equations, non-linear it can get tricky.
 

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