Fredholm integral equation needs to be written as a sum of functions
$begingroup$
solve the equation
$$ f(x) + int_0^1 (xy+x^2y^2) f(y) dy = g(x)
$$
and write in the form of
$$ sum a_jx^{j-1} $$
I have tried integration by parts but it doesn't seem to work because of f(y).
Do you need to know what g(x) is to solve this equation?
Any assistance on the method will be much appreciated.
integration definite-integrals summation
asked Dec 9 '18 at 17:02
p sp s
358
$endgroup$
add a comment |
$begingroup$
solve the equation
$$ f(x) + int_0^1 (xy+x^2y^2) f(y) dy = g(x)
$$
and write in the form of
$$ sum a_jx^{j-1} $$
I have tried integration by parts but it doesn't seem to work because of f(y).
Do you need to know what g(x) is to solve this equation?
Any assistance on the method will be much appreciated.
integration definite-integrals summation
asked Dec 9 '18 at 17:02
p sp s
358
$endgroup$
add a comment |
$begingroup$
solve the equation
$$ f(x) + int_0^1 (xy+x^2y^2) f(y) dy = g(x)
$$
and write in the form of
$$ sum a_jx^{j-1} $$
I have tried integration by parts but it doesn't seem to work because of f(y).
Do you need to know what g(x) is to solve this equation?
Any assistance on the method will be much appreciated.
integration definite-integrals summation
asked Dec 9 '18 at 17:02
p sp s
358
$endgroup$
solve the equation
$$ f(x) + int_0^1 (xy+x^2y^2) f(y) dy = g(x)
$$
and write in the form of
$$ sum a_jx^{j-1} $$
I have tried integration by parts but it doesn't seem to work because of f(y).
Do you need to know what g(x) is to solve this equation?
Any assistance on the method will be much appreciated.
integration definite-integrals summation
integration definite-integrals summation
asked Dec 9 '18 at 17:02
p sp s
358
asked Dec 9 '18 at 17:02
p sp s
358
edited Dec 9 '18 at 17:09
p s
asked Dec 9 '18 at 17:02
p sp s
358
asked Dec 9 '18 at 17:02
p sp s
358
asked Dec 9 '18 at 17:02
p sp s
358
358
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
$int_0^1 (x y^2 + x^2 y^2) f(y); dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$.
Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and
solve for $s$ and $t$ in terms of $int_0^1 y g(y); dy$ and $int_0^1 y^2 g(y); dy$.
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
$endgroup$
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$int_0^1 (x y^2 + x^2 y^2) f(y); dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$.
Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and
solve for $s$ and $t$ in terms of $int_0^1 y g(y); dy$ and $int_0^1 y^2 g(y); dy$.
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
$endgroup$
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
add a comment |
$begingroup$
$int_0^1 (x y^2 + x^2 y^2) f(y); dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$.
Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and
solve for $s$ and $t$ in terms of $int_0^1 y g(y); dy$ and $int_0^1 y^2 g(y); dy$.
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
$endgroup$
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
add a comment |
$begingroup$
$int_0^1 (x y^2 + x^2 y^2) f(y); dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$.
Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and
solve for $s$ and $t$ in terms of $int_0^1 y g(y); dy$ and $int_0^1 y^2 g(y); dy$.
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
$endgroup$
$int_0^1 (x y^2 + x^2 y^2) f(y); dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$.
Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and
solve for $s$ and $t$ in terms of $int_0^1 y g(y); dy$ and $int_0^1 y^2 g(y); dy$.
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
answered Dec 9 '18 at 17:20
Robert IsraelRobert Israel
323k23213467
323k23213467
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
add a comment |
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
How do i integrate $$ int yg(y) dy $$
$endgroup$
– p s
Dec 9 '18 at 20:46
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
$begingroup$
You don't, if you don't know what $g$ is.
$endgroup$
– Robert Israel
Dec 10 '18 at 2:50
add a comment |
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