Question

What is the solution of the integral equation g(s) = s+ ∫su2g(u) du?

Answer 1

Solution:

The given integral equation is :

$g(s)= s + \int {s u^2g (u)} \, du\\\\ = g(s)= s + s\int { u^2g (u)} \, du$------(1)

As, there is one variable to get the solution we have to differentiate it once.

g'(s)= 1 +$1\times\int { u^2g (u)} \, du$--------(2)

Writing equation 1 as :

$\frac{g(s)}{s}-1=\int { u^2g (u)} \, du$

Substituting the value of $1\times\int { u^2g (u)} \, du$ in equation ,(2)

g'(s)= 1 + $\frac{g(s)}{s}-1$

g'(s)=  $\frac{g(s)}{s}$

Which is the required solution of integral equation.