Solving ODE with Data Points: Finding Equation and Integrating Method

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SUMMARY

The discussion centers on solving the ordinary differential equation (ODE) h' = ah^b - ch^d, where b < 0, using given data points (hi, ti). The challenge lies in determining the constants a, b, c, and d to fit the model accurately. Participants suggest utilizing methods for estimating derivatives from discrete data, which can be approached as a non-linear fitting problem. Tools like Wolfram Alpha can assist in integration, but users should be prepared to work with complex functions such as the hypergeometric function.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with non-linear fitting techniques
  • Knowledge of polynomial approximation methods
  • Basic proficiency in using computational tools like Wolfram Alpha
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  • Research methods for estimating derivatives from discrete data sets
  • Explore non-linear fitting algorithms and software
  • Learn about hypergeometric functions and their applications
  • Investigate polynomial approximation techniques for ODE solutions
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Mathematicians, data scientists, and engineers working with differential equations and data modeling who need to integrate ODEs using empirical data points.

RagincajunLA
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hey guys, i was given some data points and i had to find an equation to fit the model. now my differential equation is h'= ah^b - ch^d with b < 0. I can't find any method for integrating because i don't know the constants in the equation. but i have the data points so that must help somehow. i also know the maximum and minimum of the data points. someone please help me figure out how to integrate this.
 
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If your task is to find constants a, b, c, and d, such that h(t) satisfying dh/dt = ah^b - ch^d fits the given points (hi, ti), then finding general solution to h(t) might be a backwards way of doing things.

There exist methods for finding a derivative of a function at a point given some discrete data set. These are approximate, but they essentially assume that your data is a high-order polynomial, and the solution to your ODE can be at least approximated with one.

So, use your set of hi and ti to find estimates for h'i. Now you have a standard non-linear fitting problem f(hi,a,b,c,d) = h'i. There are a number of ready algorithms and programs that can take care of both steps.
 
Otherwise if you are daring you can try wolframalpha.com [integrate 1/(a*x^b - c*x^d)]. Spoiler: you'll have to like the hypergeometric function
 

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