I have started answering a fredholm integral equation of the second kind and do not know where to go from here.

The answer has to be written in the form $$ \sum a_jx^{j-1} $$

The fredholm integral equation is

$$ x^3+\frac16x^2+\frac15x = g(x) + \lambda \int_0^1(x^2y+xy^2) f(y) dy$$.

My method so far:

Let: $$C_1 = \int_0^1yf(y)dy$$ and $$C_2 = \int_0^1y^2f(y)dy$$

Then $$ x^3+\frac16x^2+\frac15x = \lambda(C_1x^2 +C_2x) + g(x)$$.

Eliminating f(y) to get $$C_1 = \lambda(\frac14C_1 + \frac13C_2) + \int_0^1yg(y)dy$$ and $$C_2 = \lambda(\frac15C_1 + \frac14C_2) + \int_0^1y^2g(y)dy$$

I don't know where to go from here to get it into the form $$ \sum a_jx^{j-1} $$ If I have gotten anything wrong here please let me know. Any help will be appreciated

Thank you very much


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