Showing a twice differentiable function is a vector space

[vcb003104]

Homework Statement

Show that the set of twice differentiable functions f: R→R satisfying the differential equation

sin(x)f"(x)+x^{2}f(x)=0is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f.

Homework Equations

sin(x)f"(x)+x^{2}f(x)=0

The Attempt at a Solution

I have no idea how to start this question.
I know that to prove it is a vector space, proving that it is a subspace is enough?
I flipped through the notes for my course and followed an example done in class. I got :

0(x) = sin(x)0"(x) + x^{2}0(x) = 0

 

[Mark44]

Homework Statement

Show that the set of twice differentiable functions f: R→R satisfying the differential equation

sin(x)f"(x)+x^{2}f(x)=0is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f.

Homework Equations

sin(x)f"(x)+x^{2}f(x)=0

The Attempt at a Solution

I have no idea how to start this question.
I know that to prove it is a vector space, proving that it is a subspace is enough?

If your set was contained within an encompassing vector space. Can you establish that the space that your subset belongs to is a vector space? If you can't do that, you'll need to verify that all of the vector space axioms are satisfied.

I flipped through the notes for my course and followed an example done in class. I got :

0(x) = sin(x)0"(x) + x^{2}0(x) = 0

This is pretty sketchy. What are you trying to say here?

 

[vcb003104]

Hi, thanks for the quick reply.

I copied that from another example that my lecturer did in class.

I know that I need to prove the axioms but I don't know how to?

For example, to prove u+v is in V, how do I do it? what's u and what's v?

Do I say something like:
sin(u)f"(u)+u^{2}f(u)=0
sin(v)f"(v)+v^{2}f(v)=0

since u+v\in R
sin(u+v)f"(u+v)+(u+v)^{2}f(u+v)=0thanks

 

[vcb003104]

If your set was contained within an encompassing vector space. Can you establish that the space that your subset belongs to is a vector space? If you can't do that, you'll need to verify that all of the vector space axioms are satisfied.

This is pretty sketchy. What are you trying to say here?

So I googled how to prove the u + v is an element of V thing.

In this context, if I do: sin(u+v)f"(u+v) + (u+v)^{2}f(u+v) = o
I want to get it to become something like

sin(u)f"(u) + u^{2}f(u) + sin(v)f"(v)+v^{2}f(v) = 0
so 0 + 0 = 0 right?

but how can I expand them? I mean with sin(u+v) I know I can at least get sin(u)cos(v) + sin(v)cos(u) but what about f(u+v) and f"(u+v)??

Thanks a bunch
and hope I make sense

 

[HallsofIvy]

No. The point is NOT to replace "x" with u+ v. You want to prove that the set of all solutions to this (linear, homogeneous) differential equation forms a vector space. So need to show that if f(x) and g(x) both satisfy the equation then so does af(x)+ bg(x) for any numbers a and b.

 

[vcb003104]

No. The point is NOT to replace "x" with u+ v. You want to prove that the set of all solutions to this (linear, homogeneous) differential equation forms a vector space. So need to show that if f(x) and g(x) both satisfy the equation then so does af(x)+ bg(x) for any numbers a and b.

Hi there,
Thanks for the reply,

So to show f(x) and g(x) both satisfy the equation, do I just write

sin(x)f"(x)+x^{2}f(x)=0
sin(x)g"(x)+x^{2}g(X)=0


Thus af(x) + bg(x)
=a[sin(x)f"(x)+x^{2}f(x)] + b[sin(x)g"(x)+x^{2}g(X)]
=a(0) +b(0)
=0


Sorry I'm a bit slow at this

 

[HallsofIvy]

Hi there,
Thanks for the reply,

So to show f(x) and g(x) both satisfy the equation, do I just write

sin(x)f"(x)+x^{2}f(x)=0
sin(x)g"(x)+x^{2}g(X)=0Thus af(x) + bg(x)
=a[sin(x)f"(x)+x^{2}f(x)] + b[sin(x)g"(x)+x^{2}g(X)]
=a(0) +b(0)
=0

Yes, that shows that applying the differential operator to both af and bg separately gives 0.
But what you want to show is that
sin(x)(af+ bg)''+ x^2(af+ bg)= 0

Can you do that? (It is important that this is a "linear, homogeneous" differential equation and that differentiation is a "linear" operation.)

Sorry I'm a bit slow at this

 

[vcb003104]

Yes, that shows that applying the differential operator to both af and bg separately gives 0.
But what you want to show is that
sin(x)(af+ bg)''+ x^2(af+ bg)= 0

Can you do that? (It is important that this is a "linear, homogeneous" differential equation and that differentiation is a "linear" operation.)

So I can just expand it?

Can I say, showing that this is a vector space via the addition of functions:
sin(x)(af+ bg)''+ x^2(af+ bg)= 0

LHS = sin(x)(af+ bg)''+ x^{2}(af+ bg)
= sin(x)(af)"+sin(x)(bg)" + x^{2} (af) + x^{2} (bg)
= sin(x)af"(x) + x^{2}af(x) + sin(x) bg"(x) + x^{2} bg(x)
= a[sin(x)f"(x) + x^{2}f(x)] +b[sin(x) g"(x) + x^{2} g(x)]
= 0 + 0
= 0
= RHS

 

[Dick]

So I can just expand it?

Can I say, showing that this is a vector space via the addition of functions:
sin(x)(af+ bg)''+ x^2(af+ bg)= 0

LHS = sin(x)(af+ bg)''+ x^{2}(af+ bg)
= sin(x)(af)"+sin(x)(bg)" + x^{2} (af) + x^{2} (bg)
= sin(x)af"(x) + x^{2}af(x) + sin(x) bg"(x) + x^{2} bg(x)
= a[sin(x)f"(x) + x^{2}f(x)] +b[sin(x) g"(x) + x^{2} g(x)]
= 0 + 0
= 0
= RHS

Yes, that's it.

 

[vcb003104]

So that is the additional of functions right?

How about multiplication? is it included in the equation as well?

 

[Dick]

So that is the additional of functions right?

How about multiplication? is it included in the equation as well?

Yes. If you take the case a=1 and b=1, it tests addition on f+g. If you take the case g=0 it tests multiplication on af. Your test does both at once.