Solving Integral Equation: sin(x)+∫_0^π sin(x-t)y(t)dt

Click For Summary
SUMMARY

The forum discussion focuses on solving the integral equation y(x) = sin(x) + ∫0π sin(x-t)y(t)dt. Participants explore various methods, including differentiation and integration by parts, to derive a solution. The final solution is expressed as y(x) = Acos(x) + Bsin(x), with constants A and B determined through initial conditions. The discussion also touches on the application of Laplace transforms and the classification of the integral equation as a Volterra or Fredholm type.

PREREQUISITES
  • Understanding of integral equations, specifically Volterra and Fredholm types.
  • Familiarity with differentiation and integration techniques, including integration by parts.
  • Knowledge of Laplace transforms and their application in solving differential equations.
  • Basic concepts of ordinary differential equations (ODEs) and boundary conditions.
NEXT STEPS
  • Study the properties and solutions of Volterra integral equations.
  • Learn advanced techniques in integration by parts for complex integrals.
  • Explore the application of Laplace transforms in solving integral and differential equations.
  • Investigate the relationship between integral equations and ordinary differential equations.
USEFUL FOR

Mathematicians, students studying integral equations, and anyone interested in advanced calculus and differential equations will benefit from this discussion.

  • #31
It's not Ax+B, it's f(x)=Aln(x)+B. You want to integrate ln(x/t)*f(t)=ln(x/t)*(Aln(t)+B)dt from t from 0 to 1. That breaks up into integrating ln(t)^2 and ln(t). Do them by integrating by parts. If you are having trouble with the individual integrations, it might be best to post a separate thread with your specific integration questions. This doesn't have much to do with integral equations.
 
Physics news on Phys.org
  • #32
Have you tried the laplace transform ... it applies very nicely here .:)

This equation is the Volterra integral equations .
 
  • #33
How does the Laplace trandform apply here?

Which integral equaiton is Volterra?
The ones are posted are Fredholm.
 

Similar threads

Calculating radius of gyration of plane figure about x-axis
  • · Replies 5 ·
Replies
5
Views
2K
Determine limits of integration in double integral change of variables
  • · Replies 2 ·
Replies
2
Views
2K
How should I use the averaging approximation to find this?
  • · Replies 3 ·
Replies
3
Views
2K
Area surface of revolution (rotating this astroid curve around the x-axis)
  • · Replies 2 ·
Replies
2
Views
2K
Solve the problem involving the given double integral
  • · Replies 1 ·
Replies
1
Views
2K
Solving the wave equation with piecewise initial conditions
  • · Replies 11 ·
Replies
11
Views
3K
Curve for a line integral - direction confusion
  • · Replies 9 ·
Replies
9
Views
1K
Avoid unpleasant integrals in solving IVP
  • · Replies 3 ·
Replies
3
Views
2K
Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
  • · Replies 7 ·
Replies
7
Views
2K
Is the Calculation of the Vector Line Integral Over a Square Correct?
  • · Replies 12 ·
Replies
12
Views
2K