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In summary, the differential equation y''-xy=0 is being solved using power series, with a given initial condition of x0 = -1. The equation is set up as Ʃ(n+2)(n+1)an+2(x+1)n - x Ʃ an(x+1)n. The speaker is familiar with solving equations like this, but has not encountered one where the x needs to be distributed. They use the trick xƩan(x+1)n = (x+1-1)Ʃan(x+1)n and distribute the x+1 to obtain -Ʃan(x+1)n+1. They ask for someone to double check their solution, and it is confirmed
  • #1
twisted079
25
1
So the differential equation I have to solve using power series is
y''-xy=0 when x0 = -1

So i set it up
Ʃ(n+2)(n+1)an+2(x+1)n - x Ʃ an(x+1)n

I know how to generally solve equations like this, but I never solved one like this, where I have to distribute the x ... x(x+1)n ... I just need to figure this part out (I know I left out the n=0 to ∞)
 
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  • #2
Ok I figured it out using a cheap (but valid) math trick... in case anyone is wondering...

xƩan(x+1)n = (x+1-1)Ʃan(x+1)n

Now the x+1 can be distributed to give -Ʃan(x+1)n+1

...anyone care to double check me on this?
 
  • #3
twisted079 said:
Ok I figured it out using a cheap (but valid) math trick... in case anyone is wondering...

xƩan(x+1)n = (x+1-1)Ʃan(x+1)n

Now the x+1 can be distributed to give -Ʃan(x+1)n+1

...anyone care to double check me on this?

Yep, that's the trick you want to use.
 

1. How do I solve a series quickly?

One way to solve a series quickly is to use a specific formula or pattern to find the sum, rather than adding each individual term. Another method is to use a calculator or computer program to calculate the sum.

2. What is the quickest way to find the sum of a series?

The quickest way to find the sum of a series is to use a specific formula or pattern to calculate the sum, rather than adding each individual term. This method is more efficient and saves time compared to manually adding each term.

3. Can you provide an example of finding the sum of a series quickly?

For example, if you have the series 1 + 2 + 3 + 4 + 5, you can use the formula n(n+1)/2, where n is the number of terms, to find the sum. In this case, n = 5, so the sum is (5)(5+1)/2 = 15.

4. How do I know which formula or method to use for a particular series?

There are various methods and formulas for finding the sum of different types of series, such as arithmetic, geometric, or telescoping series. It is important to identify the type of series and then use the appropriate formula or method to find the sum.

5. Is there a way to check if my answer for the sum of a series is correct?

Yes, there are online calculators and computer programs that can verify the sum of a series. You can also manually check your answer by using different methods or formulas to see if you get the same result.

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