Solving a differential equation with generalized functions
$begingroup$
We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,varphi)$ and $(y',varphi) = (y,-varphi')$.
The textbook I am currently reading uses the following statements to prove that the solution to this equation is $y = C (C = const)$:
1) $varphi_0 {(x)} = varphi_1' (x)$ iff $ intlimits_{-infty}^{+infty}{varphi_0 (x)dx = 0}$
2) Given (1), we assume:
$varphi_1 (x) = intlimits_{-infty}^{x}{varphi_0 (t)dt}$
3) Now let $varphi_1 (x)$ be a fixed function with the property:
$intlimits_{-infty}^{+infty}varphi_1(x)dx = 1$
4) For every $varphi(x)$ we can write the following equality:
$varphi(x) = varphi_1(x)intlimits_{-infty}^{+infty}varphi(x)dx + varphi_0 (x)$ where $varphi_0(x)$, obviously, satisfies (1).
What I can't understand is the fourth equation. How is it derived? The equality looks as if integration by parts was used, but I can't get the same result.
What am I missing?
analysis mathematical-physics
asked Jan 21 at 13:29
Don DraperDon Draper
87110
$endgroup$
add a comment |
$begingroup$
We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,varphi)$ and $(y',varphi) = (y,-varphi')$.
The textbook I am currently reading uses the following statements to prove that the solution to this equation is $y = C (C = const)$:
1) $varphi_0 {(x)} = varphi_1' (x)$ iff $ intlimits_{-infty}^{+infty}{varphi_0 (x)dx = 0}$
2) Given (1), we assume:
$varphi_1 (x) = intlimits_{-infty}^{x}{varphi_0 (t)dt}$
3) Now let $varphi_1 (x)$ be a fixed function with the property:
$intlimits_{-infty}^{+infty}varphi_1(x)dx = 1$
4) For every $varphi(x)$ we can write the following equality:
$varphi(x) = varphi_1(x)intlimits_{-infty}^{+infty}varphi(x)dx + varphi_0 (x)$ where $varphi_0(x)$, obviously, satisfies (1).
What I can't understand is the fourth equation. How is it derived? The equality looks as if integration by parts was used, but I can't get the same result.
What am I missing?
analysis mathematical-physics
asked Jan 21 at 13:29
Don DraperDon Draper
87110
$endgroup$
add a comment |
$begingroup$
We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,varphi)$ and $(y',varphi) = (y,-varphi')$.
The textbook I am currently reading uses the following statements to prove that the solution to this equation is $y = C (C = const)$:
1) $varphi_0 {(x)} = varphi_1' (x)$ iff $ intlimits_{-infty}^{+infty}{varphi_0 (x)dx = 0}$
2) Given (1), we assume:
$varphi_1 (x) = intlimits_{-infty}^{x}{varphi_0 (t)dt}$
3) Now let $varphi_1 (x)$ be a fixed function with the property:
$intlimits_{-infty}^{+infty}varphi_1(x)dx = 1$
4) For every $varphi(x)$ we can write the following equality:
$varphi(x) = varphi_1(x)intlimits_{-infty}^{+infty}varphi(x)dx + varphi_0 (x)$ where $varphi_0(x)$, obviously, satisfies (1).
What I can't understand is the fourth equation. How is it derived? The equality looks as if integration by parts was used, but I can't get the same result.
What am I missing?
analysis mathematical-physics
asked Jan 21 at 13:29
Don DraperDon Draper
87110
$endgroup$
We need to solve the equation $y' = 0$ (for the class of generalized functions where $y = (y,varphi)$ and $(y',varphi) = (y,-varphi')$.
The textbook I am currently reading uses the following statements to prove that the solution to this equation is $y = C (C = const)$:
1) $varphi_0 {(x)} = varphi_1' (x)$ iff $ intlimits_{-infty}^{+infty}{varphi_0 (x)dx = 0}$
2) Given (1), we assume:
$varphi_1 (x) = intlimits_{-infty}^{x}{varphi_0 (t)dt}$
3) Now let $varphi_1 (x)$ be a fixed function with the property:
$intlimits_{-infty}^{+infty}varphi_1(x)dx = 1$
4) For every $varphi(x)$ we can write the following equality:
$varphi(x) = varphi_1(x)intlimits_{-infty}^{+infty}varphi(x)dx + varphi_0 (x)$ where $varphi_0(x)$, obviously, satisfies (1).
What I can't understand is the fourth equation. How is it derived? The equality looks as if integration by parts was used, but I can't get the same result.
What am I missing?
analysis mathematical-physics
analysis mathematical-physics
asked Jan 21 at 13:29
Don DraperDon Draper
87110
asked Jan 21 at 13:29
Don DraperDon Draper
87110
edited Jan 21 at 13:34
Don Draper
asked Jan 21 at 13:29
Don DraperDon Draper
87110
asked Jan 21 at 13:29
Don DraperDon Draper
87110
asked Jan 21 at 13:29
Don DraperDon Draper
87110
87110
add a comment |
add a comment |
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