Reformulating Integrals as ODE Problems

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Discussion Overview

The discussion revolves around the process of reformulating integrals as ordinary differential equation (ODE) problems. Participants explore the conditions under which this reformulation is possible, particularly in the context of definite and indefinite integrals, and the implications for numerical methods in MATLAB.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to reformulate an integral as an ODE problem.
  • Another suggests differentiating both sides of the equation but questions if that is the correct approach.
  • Concerns are raised about the feasibility of reformulating a definite integral, with examples provided to illustrate the distinction between definite and indefinite integrals.
  • Participants discuss the relationship between the integral of a function m(x, y) and the resulting function M(y) after integration with respect to x.
  • There is a proposal that the only way to reformulate the integral is to differentiate it with respect to y.
  • One participant expresses a desire to rewrite the integral as an ODE without needing to solve the integral itself.
  • Another participant questions whether the integral can be associated with an ODE, suggesting that the integral yields a number rather than a function.
  • Clarification is provided that y is not a function of x, and the context of the problem is related to numerical analysis using MATLAB.
  • Participants discuss the use of MATLAB functions like quadl and ode45, questioning the relationship between their outputs.
  • There is confusion regarding the initial value used in the ode45 function and its expected outcome compared to the quadl function.

Areas of Agreement / Disagreement

Participants express differing views on the possibility of reformulating definite integrals as ODE problems. There is no consensus on the correct approach or the relationship between the numerical methods discussed.

Contextual Notes

Participants highlight limitations in understanding the relationship between the integral and its reformulation as an ODE, particularly regarding the definitions and assumptions about the variables involved.

Who May Find This Useful

This discussion may be useful for students and practitioners in numerical analysis, particularly those interested in the application of MATLAB for solving integrals and differential equations.

JohanL
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How do you reformulate an integral as an ode-problem?
 
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Differentiate both sides of the equation? Or do you mean something else? :/
 
But is that possible if you have an integral between two values. I know its possible if you have an indefinite?! integral...now i have

M(y)=int(a,b) m(x,y) dx
 
JohanL said:
But is that possible if you have an integral between two values. I know its possible if you have an indefinite?! integral...now i have

M(y)=int(a,b) m(x,y) dx

m(x, y) is a function of x and y if you integrate this with respect to x you will get a function of y, so that will then be M(y):

[tex]\int_a^b m(x, y) dx = [f(x, y)]_a^b = f(b, y) - f(a, y) = M(y)[/tex]
 
But I want to reformulate this integral to an ode-problem
 
well I think the only way you can do that is like Nylex said, like this:

[tex]\frac{d(\int_a^b m(x, y) dx)}{dy} = \frac{dM(y)}{dy}[/tex]
:redface:
 
But with this I still need to solve the integral, don't I?
And I want to rewrite it as an ode so i don't have to solve the integral.
 
From your posts it isn't clear if y is a function of x or not, since it usually isn't that the integral has limits a and b.

An integral that can be turned into a differential (ODE) would be something like:


[tex]\int_0^xf(t)dt = y[/tex]

which has associated differential equation dy/dx = f(x)


so why do you even think that the type of equation you wrote has an ODE equivalent?

Even if y were a function of x, then the integral you wrote would still only yield a number, and that isn't the function y.
 
y isn't a function of x. a and b are reell numbers.
The question in the book is(numerical analysis)

M(y)=int(a,b) m(x,y) dx
a. Use quadl in MATLAB to determine M
b. An alternative way to determine M is to rewrite the integral as an ODE-problem. Do that and use ode45 in MATLAB to solve the ode-problem. Compare with the solution in a.

b is really strange...
 
  • #10
what ode45(f, [a, b], y0) does is simply integrating f from a to b, using inital value y0...

so you can use that to get your answer (you will need to use the @ again ;-), like this ode45(@f, [a, b], 0)
 
  • #11
Do you mean that the initial value is f(a). Then
ode45(@f, [a, b], y0)
should yield the same value as
QUADL(@f,a,b);
but it dont.
I want them to yield the same value...what am i doing wrong...
 
  • #12
I think the inital value should be f(0)
 

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