Solving Integral Equation: \int_0^2 t y(t)dt = 3

[sara_87]

Homework Statement

solve the integral equation:

$$\int_0^2 t y(t)dt = 3$$

Homework Equations

The Attempt at a Solution

If we differentiate we get:
0 = xy(x)
which means y=0.
but this is the wrong answer and i think this is the wrong approach.

 

[torquil]

Your proposed calculation is not correct (as you know of course), because both sides of the integral equation are simply constant numbers, i.e. they don't depend on any variable with respect to which you can differentiate.

It would have been correct if the upper integration limit had been t, but here it is simply 2.

So you must think a bit differently on this one.

Torquil

 

[sara_87]

i can't think of a different method
:(

 

[chrispb]

Something about that problem statement appears to be suspicious... Solving an integral equation would normally mean solving for the general form of y(x), and applying boundary conditions to get it into the exact form you want. Here, there aren't enough constraints to uniquely specify y; in fact, there aren't even enough constraints to specify the FORM of y! You could write y = C, y = Ct, y = Csin(t), y = Ce^t, etc, and then solve for C in each of those by doing the integral to get different expressions for y. In short, this problem makes no sense.

 

[torquil]

Ok, but it's not really "a method".

Hint: There is an infinite number of different functions y(t) that will be able to solve the equation. In fact, almost any integrable function will be able to do it, if it is correctly normalized.

The problem only says to "solve it", so any solution will do.

Torquil

 

[sara_87]

oh right, i see.
like y=3/2 can be a solution for an example.

so there's infinitely many solutions.

 

[torquil]

Almost...

EDIT: Yes! :-)

 

[sara_87]

thanks
:)