# solution of integral equation

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• Nov 29th 2013, 07:48 AM
Suvadip
solution of integral equation
I have to solve the integral equation $\displaystyle y(x)=1+2\int_0^x(t+y(t))dt, y_0(x)=1$ by the
method of successive approximation. Using $\displaystyle y_n(x)=1+x^2+2\int_0^x y_{n-1}(t)dt$, I can find $\displaystyle y_1(x), y_2(x),$ etc
but I cannot find any general expression for $\displaystyle y_n(x)$ and hence cant find the solution $\displaystyle y(x)(=\mbox{Lim} y_n(x))$ . Please help
• Nov 29th 2013, 03:42 PM
romsek
Re: solution of integral equation
it's there

take the polynomial out to x8 or so and look at the first 7 terms, ignore the 8th

you should see 1 + x + 3/2x2 + 1/2x3 + 1/8x4 + ....

you should see this as 1 + x + Sum[2,Infinity, some pattern]

The sum will correspond to a well known power series minus a couple terms. You're also adding a couple terms at the front.

Collect all that together and you'll end up with an expression for the solution as a function of x.