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Math10 said:Homework Statement
Find the power series in x-x_{0} for the general solution of y"-y=0; x_{0}=3.
Homework Equations
None.
The Attempt at a Solution
Let me post my whole work:
Math10 said:Can you please take a look at the work that I posted? It's clearly written.
Math10 said:I know that's the right answer, but what should I do to get to the right answer after the last step in my work? That's where I got stucked.
Math10 said:You mean this:
n=2m (even index)
a_{2m+2}=a_{2m}/[(2m+2)(2m+1)]
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n=2m+1 (odd index)
a_{2m+3}=a_{2m+1}/[(2m+3)(2m+2)]
Math10 said:So how do I get to the answer? I know where x-3 comes from.
Math10 said:I still don't really get it.
This thread is in the Homework Help forums...PeroK said:You have:
##(n+2)(n+1)a_{n+2} = a_n##
Hence:
##a_{n+2} = \frac{a_n}{(n+2)(n+1)}##
For ##n## even this gives:
##a_2 = \frac{a_0}{2}, \ a_4 = \frac{a_2}{12} = \frac{a_0}{24}, \ a_6 = \frac{a_4}{30} = \frac{a_0}{720} \dots##
And, now by insight, inspiration (or looking at the answer) you have to notice that ##2, 24, 720 \dots## are the even factorials and hence ##a_n = \frac{a_0}{n!}##
Odd ##n## is much the same.
A power series in mathematics is an infinite series of the form ∑_{n=0}∞ c_{n}(x-x_{0})^{n}, where c_{n} are coefficients and x_{0} is a constant. It is used to represent a function as a sum of infinitely many terms involving powers of (x-x_{0}).
Power series are useful in scientific research because they can be used to approximate functions that are difficult to evaluate or solve analytically. They are also used in many branches of mathematics, such as calculus and differential equations.
The process for finding the power series in x-x_{0} involves expanding the function into an infinite series by using the Taylor series expansion formula and then simplifying the terms to match the format of a power series. The coefficients can then be calculated using various methods, such as differentiation or integration.
Power series have many applications in real-world problems, such as in physics, engineering, and economics. They are used to model and approximate various phenomena, such as motion, heat transfer, and population growth.
There are some limitations to using power series in scientific calculations. They can only approximate functions within a certain radius of convergence, and the accuracy of the approximation decreases as you move further away from the center point. Additionally, power series may not converge for all values of x, making them unsuitable for certain types of functions.