Finding the coefficients of an unknown function using Taylor's Theorem?

[SMA_01]

Homework Statement

Find coefficients A, B, and C.
f'(x)= Af(x)+Bf(x+h)+Cf(x+2h)+O(h²)
Using Taylor's Theorem.

Note: O stands for Big O in asymptotic order notation.

The Attempt at a Solution

Here are the expansions:

Bf(x+h)= Bf(x)+Bhf'(x)+(1/2)Bh²f"(x)+(1/6)Bh³f"'(x)...

Cf(x+2h)=Cf(x)+2Chf'(x)+2Ch²f"(x)+(4/3)Ch³f"'(X)...

And then I added them and factored out the coefficients

= (A+B+C)f(x)+(B+2C)hf'(x)+(1/2B+2C)h²f"(x)+...

Is this correct? I'm stuck as to what I am supposed to do next.

Thanks.

 

[Dick]

Homework Statement

Find coefficients A, B, and C.
f'(x)= Af(x)+Bf(x+h)+Cf(x+2h)+O(h²)
Using Taylor's Theorem.

Note: O stands for Big O in asymptotic order notation.

The Attempt at a Solution

Here are the expansions:

Bf(x+h)= Bf(x)+Bhf'(x)+(1/2)Bh²f"(x)+(1/6)Bh³f"'(x)...

Cf(x+2h)=Cf(x)+2Chf'(x)+2Ch²f"(x)+(4/3)Ch³f"'(X)...

And then I added them and factored out the coefficients

= (A+B+C)f(x)+(B+2C)hf'(x)+(1/2B+2C)h²f"(x)+...

Is this correct? I'm stuck as to what I am supposed to do next.

Thanks.

That looks pretty odd. I think there is a typo in the problem. Try changing the left hand side from f'(x) to f'(x)*h. Now it makes a little more sense. Try it from there.