Verifying a Power Series Solution for y''-4y=0

In summary, a power series is an infinite series in the form of ∑<sub>n=0</sub>∞ a<sub>n</sub>(x-c)<sup>n</sup>, where a<sub>n</sub> are constants and x is the variable. Its derivative can be obtained by differentiating each term, resulting in a new series with the coefficients multiplied by their respective exponent and the exponent reduced by 1. This can be calculated using the rules of differentiation, but it's important to note that the interval of convergence may differ. A power series can be differentiated term by term if it is convergent, and this has significance in solving optimization problems and approximating function values within the interval of convergence.
  • #1
joker2014
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Homework Statement


substitute the given power series below into ODE y'' -4y=0 to verify it is a solution

Homework Equations


y=∑ 2n xn / n!
n=0

y''-4y=0

The Attempt at a Solution



I have absolutely no idea how start.
 
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  • #2
you must know the derivative of ##x^n## by now. that's somewhere you could start.
 
  • #3
Locked because no attempt was shown.
 

1. What is a power series?

A power series is an infinite series of the form ∑n=0∞ an(x-c)n, where an are constants and x is the variable. It is a type of Taylor series that can be used to represent a function as a polynomial, with c being the center of the series.

2. What is the derivative of a power series?

The derivative of a power series is obtained by differentiating each term in the series individually. This results in a new power series with the coefficients multiplied by their respective exponent and the exponent reduced by 1.

3. How is the derivative of a power series calculated?

The derivative of a power series can be calculated using the rules of differentiation, such as the product rule and chain rule. It is important to note that the interval of convergence for the original power series and its derivative may differ.

4. Can a power series be differentiated term by term?

Yes, a power series can be differentiated term by term as long as the series is convergent. This means that the derivative of the power series will also be a power series with the same interval of convergence.

5. What is the significance of finding the derivative of a power series?

Finding the derivative of a power series can help in solving problems related to optimization and finding the behavior of a function. It can also be used to approximate the value of a function at a specific point within the interval of convergence.

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