Solving an integral equation by iteration

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Discussion Overview

The discussion revolves around solving a Volterra integral equation through iteration and comparing its solution to that of a corresponding differential equation. Participants explore the relationship between the two equations, particularly focusing on initial conditions and their implications for the solutions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a Volterra integral equation and expresses uncertainty about how to begin solving it through iteration.
  • Another participant questions the need for different initial conditions when comparing solutions to the same initial value problem.
  • A participant argues that iterating the integral equation and solving the differential equation using series leads to a discrepancy, specifically the absence of a term (1+x) in the integral solution.
  • One participant asserts that the initial conditions provided in the problem statement are incorrect, suggesting they should be y(0)=y'(0)=0 instead of y(0)=y'(0)=1.
  • A later reply supports the assertion that the professor's claim about the initial conditions being correct may be mistaken.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of the initial conditions for the differential equation and the integral equation. There is no consensus on the resolution of the initial conditions issue, as some participants believe the professor is incorrect while others raise questions about the implications of changing initial conditions.

Contextual Notes

Participants note that the integral equation leads to different initial conditions than those provided for the differential equation, which complicates the comparison of solutions. The discussion highlights the need for clarity on the relationship between the two equations and their respective initial conditions.

mode1111
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Hello guys!

I was given a Volterra integral equation y(x)=1/2*x^2+integral(0--->x) [t(t-x)y(t)]dt to solve using iteration. I have no idea how and where to start...

The full problem goes as follows:
Show that the solution y(x) of y''+xy=1, y(0)=y'(0)=1 also satisfies the integral equation (above). Use iteration to solve the integral equation, and compare with the series solution of the differential equation.

The series solution (done with taylor) equals: 1+x+1/2*x^2-1/2*x^3-1/6*x^4-...
when you turn the differential equation to an integral equation it looks the same except it has additional 1+x that come from the initial conditions. So basically, to compare them we need different initial conditions i.e y(0)=y'(0)=0 and not 1. However I've asked my professor, and he said there is no such problem and that the algebra was wrong. I can't find the error...

Please help me figure this out.
Thanks to anyone that tries!
 
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I don't understand why you would need different initial conditions. You are comparing different solutions to the same initial value problem aren't you? Changing the initial conditions would change the problem.

And why do you want to "turn the differential equation to an integral equation"? You are given the integral equation to begin with.
 
I totally agree with you, however if you try to iterate the integral equation and to solve the differential one using series solution, you'll find that you lack the 1+x part. This fact made me question the initial conditions so i tried to make the two equations look the same. It turned out that they are indeed the same, but the conditions are different, i.e the integral one doesn't have the 1+x part that comes from y(0)=y'(0)=1. Again, I did something wrong here and I don't know what. If you could please explain or write down your solution, I'll be more than grateful to you...
Thank you.
 
mode1111 said:
Hello guys!

I was given a Volterra integral equation y(x)=1/2*x^2+integral(0--->x) [t(t-x)y(t)]dt to solve using iteration. I have no idea how and where to start...

The full problem goes as follows:
Show that the solution y(x) of y''+xy=1, y(0)=y'(0)=1 also satisfies the integral equation (above). Use iteration to solve the integral equation
Well, this can't be right. From the integral equation, [itex]y(0)= (1/2)(0)+ \int_0^0 [t^2y(t)dt= 0[/itex], not 1. Further, [itex]y'(x)= x- \int_0^x ty(t)dt[/itex] so that [itex]y(0)= 0- \int_0^0 ty(t)dt= 0[/itex]. The differential equation is correct but the initial values are wrong. They should be y(0)= y'(0)= 0.
 
Yeah, that is exactly my point...I guess the my professor was wrong after all.
Thanks, HallsofIvy!
 

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