Differential equations - algebraic properties of solutions

[braindead101]

Show that the operator A defined by A[y](t) = integ(s(y(s))ds) [integration from t to a] is linear. (Hint: Use integration by parts.)



Attempt at the solution:
A[y](t) = integ(s(y(s))ds) [integ from t to a]

Using integration by parts:
Let u = s
du = ds

let dv = y(s)
v = 1/2 y^2(s)

so,
A[y](t) = s(1/2y^2(s)) - integ(ds 1/2y^2(s)) [i'm not sure how to change the integration limits]
A[y](t) = 1/2s y^2(s) - [1/2 (1/3 y^3(s))] (from ___ to ___)
and just sub in the integrating limits and simplify?
am i doing something wrong.

 

[cristo]

Show that the operator A defined by A[y](t) = integ(s(y(s))ds) [integration from t to a] is linear. (Hint: Use integration by parts.)

Well, firstly, what is the definition of a linear operator?

Attempt at the solution:
A[y](t) = integ(s(y(s))ds) [integ from t to a]

Does this mean s times the function y(s)? i.e. the integral $$\int_t^asy(s)ds$$.

Using integration by parts:
Let u = s
du = ds

let dv = y(s)
v = 1/2 y^2(s)

y is a function, and so you can't integrate it like this! I would let dv=s ds and u=y(s). But then, I don't know the definition of a linear operator that you are using, and so do not know what you're trying to prove!

 

[HallsofIvy]

Show that the operator A defined by A[y](t) = integ(s(y(s))ds) [integration from t to a] is linear. (Hint: Use integration by parts.)



Attempt at the solution:
A[y](t) = integ(s(y(s))ds) [integ from t to a]

Using integration by parts:
Let u = s
du = ds

let dv = y(s)
v = 1/2 y^2(s)

so,
A[y](t) = s(1/2y^2(s)) - integ(ds 1/2y^2(s)) [i'm not sure how to change the integration limits]
A[y](t) = 1/2s y^2(s) - [1/2 (1/3 y^3(s))] (from ___ to ___)
and just sub in the integrating limits and simplify?
am i doing something wrong.

What in the world are you doing? No one asked you to actually integrate!

A transformation, T, is linear if and only if T(ay₁+ by₂)= aT(y₁)+ bT(y₂).

What can you do with $\int s(ay_1+ by_2)ds$?

 

[braindead101]

i integrated since the hint was to use integration of parts
i have no idea what i am doing

 

[braindead101]

i'm looking at the integral you posted hallsofivy
you can multiply the s in and split it into two integrals?

 

[radou]

i'm looking at the integral you posted hallsofivy
you can multiply the s in and split it into two integrals?

You can, since linearity is a property of intergals.

 

[braindead101]

can i integrate it then?
or is that not the point.
im not quite sure what I'm doing, I've read the class notes. i don't really understand

 

[braindead101]

anybody know what i can do
this is relaly causing me trouble

 

[HallsofIvy]

First go back and check the problem! If you are really to "Show that the operator A defined by A[y](t) = integ(s(y(s))ds) [integration from t to a] is linear" then do as Cristo suggested originally- check the definition of "linear" and show that it works here. I can not imagine how one would "use integration by parts" to do that.

 

[braindead101]

definition of a linear operator means it must satisfy A[cy] = cA[y] and A[y1+y2]=A[y1] + A[y2]

i would probably know how to do this if it wasn't an integral and i had to show it is linear, but i really do not know where to start.

 

[braindead101]

i think i have an idea how to do it now, without the integration by parts.. but my prof said that we are suppose to do it using that, i have no idea how.

but this is what I am doing right now (it seems right to me but correct me if I'm wrong):
A[y] = integ(s(y(s))ds)

A[cy] = integ(s(cy) ds)
A[cy] = c integ(s y(s) ds)
A[cy] = c A[y]
so i satisfied property 1

A[y1 + y2] = integ( s(y1+y2) ds)
A[y1 + y2] = integ( sy1 ds) + integ( sy2 ds)
A[y1 + y2] = A[y1] + A[y2]
so i satisfied property 2

so it is a linear operator?
but since i need integration by parts, I've no idea what to do.