Fredholm integral equation - Degenerate kernel method
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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
calculus linear-algebra integration functional-analysis definite-integrals
asked Dec 29 '18 at 18:03
p sp s
378
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add a comment |
$begingroup$
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
calculus linear-algebra integration functional-analysis definite-integrals
asked Dec 29 '18 at 18:03
p sp s
378
$endgroup$
add a comment |
$begingroup$
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
calculus linear-algebra integration functional-analysis definite-integrals
asked Dec 29 '18 at 18:03
p sp s
378
$endgroup$
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
calculus linear-algebra integration functional-analysis definite-integrals
calculus linear-algebra integration functional-analysis definite-integrals
asked Dec 29 '18 at 18:03
p sp s
378
asked Dec 29 '18 at 18:03
p sp s
378
edited Jan 2 at 18:03
p s
asked Dec 29 '18 at 18:03
p sp s
378
asked Dec 29 '18 at 18:03
p sp s
378
asked Dec 29 '18 at 18:03
p sp s
378
378
add a comment |
add a comment |
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