Solving an integral equation

I'm looking to implement a solution to the following equation:

, where a is a known constant, for demonstration purposes set to 5 below.

Code that I tried:

syms r(t)
i=int(1/(r.^4),0,t)
solve(r==(1-5*i),r)

No clue how to tackle the problem. Thanks in advance!

Answer 1

[Bjorn Gustavsson]

For this I think pen-and-paper might be easier than using the symbolic toolbox. Simply differentiate both sides of your definition of r and convert your integral equation into a differential equation:

This simplifies to (at least for some nice-constraints on r):

That way you now have a reasonably ordinary ODE instead. This is most hand-wavy lacking mathematical rigor, but you should be able to add that if required.

HTH

[Kaz Middelhoek]

Thank you very much! A smart way to tackle the problem.

[Bjorn Gustavsson]

You're welcome. This is the standard way (I've been taught) of attacking this type of integral equations, for others you might start to look at the methods for solving integral equations in chebfun.