If $y(t) = t\left(1-\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to?

Asked Feb 10 '16 at 05:06

Active Feb 10 '16 at 06:03

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Leibniz rule or Laplace transform?

Let $y(t)$ be a continuous function on $[0,\infty)$. If $$y(t) = t\left(1-4\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to __________.

Source.

I solved this question using Leibniz rule and converting this Fredholm integral equation into ordinary differential equation and got my answer to be $0.5$.

First of all is it correct?
Secondly somebody said you could solve this using Laplace transformation also assuming it exists for $y$? Can be somebody show me how to do this?

edited Feb 10 '16 at 06:03

asked Feb 10 '16 at 05:06

Are you sure this is Fredholm I. E.? – Mhenni Benghorbal Feb 10 '16 at 05:21
1

I am sorry this volterra – Feb 10 '16 at 05:22
Do you have a background about Laplace transform? – Mhenni Benghorbal Feb 10 '16 at 05:34
Not much but i know how to find laplace transforms – Feb 10 '16 at 05:36
Then take the Laplace transform of the equation you w. r. to t and see here. – Mhenni Benghorbal Feb 10 '16 at 05:38
I don't understand how to take the laplace transform and convlution ?please solve my question for me the way you are saying it would be of great help – Feb 10 '16 at 05:42
That's why I asked you about your background! You need to more about Laplace transform to solve such eq. By the way your eq. is not convolution type. – Mhenni Benghorbal Feb 10 '16 at 05:46
Ohk.then what do i do – Feb 10 '16 at 05:47
As I said take some time to learn more about Laplace transform! Even if people solved it for you you would not be able to understand their solution! – Mhenni Benghorbal Feb 10 '16 at 05:53
Oh then plz solve it using lebnitz rule as i am not sure of my answer – Feb 10 '16 at 05:55
See the second approach in my answer. – Mhenni Benghorbal Feb 10 '16 at 06:10
Identical to http://math.stackexchange.com/questions/1643665/what-is-the-value-of-integral/1643695#1643695 which received an elementary solution – Carl Christian Feb 10 '16 at 13:00

If $y(t) = t\left(1-\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to?

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