Finding Values of 'a' for f ''(x) + f(x)=0

  • Thread starter oswald
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In summary, the student is trying to find Maclaurin series for A sin(ax) and B cos(ax). They find that A sin(ax) has a Maclaurin series as A[∑ (-1)^k . (ax)^2k+1 ] / (2k+1)! and B cos(ax) has a Maclaurin series as B[∑ (-1)^k . (ax)^2k ] / (2k)!. They then use these series to find a and b.
  • #1
oswald
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Let us take the first three non-zero terms of Maclaurin's expansion of
f(x)= Asen(ax) + Bcos(ax).
Determine for which values of a the equation f ' '(x) + f(x) = 0.
 
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  • #2
Welcome to Physicsforums Oswald.

In this case it seems better to leave f(x) in exact form. Can you find f''(x) and express it in terms of f(x) ?
 
  • #3
oswald said:
Let us take the first three non-zero terms of Maclaurin's expansion of
f(x)= Asen(ax) + Bcos(ax).
Determine for which values of a the equation f ' '(x) + f(x) = 0.

That looks like a homework problem and you haven't said anything about what YOU have already done. Do you know how to find the MacLaurin's series for a function?
 
  • #4
Maclaurin Serie solution

I have done this, but it doesn't work...
......∞
A sin(ax) = A[ ∑ (-1)^k . (ax)^2k+1 ] / (2k+1)! to find Maclaurin series of A sin(ax)
......0
......∞
B cos(ax) = B[ ∑ (-1)^k . (ax)^2k ] / (2k)! is Maclaurin Series of B cos(ax)
......0

hence:

A sin(ax) = Aax - Aa³x³/6 + Aa^5x^5/120

B cos(ax) = 1 - Ba²x²/2 + Ba^4x^4/24

f''(x) + f(x) = 0

what is f''(x)? I think its:

f''(x) = 0 + 1
f(x) = A sen(ax) + B cos(ax)

so,

0 + 1 + Asen(ax) + B cos(ax) = 0

Asen(ax) + Bcos(ax) = -1

the answer is a= 1 and a= -1, but i don't know how. Is the value of A, B and x important?
 
Last edited:
  • #5
Hi,i found the solution in yahoo...

Take your function and differentiate it twice. observe that

f '' (x) = - a^2 f(x)

Substitute this into the equation. One possibility is that
A=B=0. Otherwise, cancel f(x) and you have a simple algebraic equation for a to solve.
have done:
f²(x)= -a²f(x)
so,
-a²f(x)+f(x)=0
f(x)[-a²+1]=0
a= 1 or -1!
i don't know why the teacher ask for Maclaurin series...
 
  • #6
I don't know why you didn't just take my hint, which got at exactly the same thing, rather than go to Yahoo.
 

1. What is the purpose of finding values of 'a' for f ''(x) + f(x)=0?

The purpose of finding values of 'a' is to determine the possible solutions or roots of the given equation, which can help in understanding the behavior and properties of the function f(x).

2. What is the significance of the second derivative in this equation?

The second derivative, denoted by f ''(x), measures the rate of change of the first derivative f '(x). It helps in identifying the concavity or curvature of the graph of f(x) and provides information about the maximum and minimum points of the function.

3. How can we solve for 'a' in the equation f ''(x) + f(x)=0?

To solve for 'a', we can use various methods such as substitution, factoring, completing the square, or using the quadratic formula. The method used will depend on the given equation and the level of complexity.

4. Are there any restrictions on the value of 'a' in this equation?

Yes, there are certain restrictions on the value of 'a' in this equation. For example, if the equation is given in the form of a trigonometric function, then 'a' must be within a specific range to ensure that the function is well-defined. Additionally, the value of 'a' may also affect the domain and range of the function.

5. How can finding values of 'a' for f ''(x) + f(x)=0 be useful in real-life applications?

Finding values of 'a' can be useful in various fields such as physics, engineering, and economics. For instance, in physics, it can help in determining the conditions for an object to be in equilibrium. In economics, it can help in analyzing the stability of a market. Overall, understanding the behavior of a function through values of 'a' can provide insights into real-life situations and aid in decision-making processes.

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