Find the power series in x for the general solution of?

1. Jul 27, 2016

Math10

1. The problem statement, all variables and given/known data
Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0.

2. Relevant equations
None.

3. The attempt at a solution
I got up to an+2=-an(n+3)/(n+1)
for n=1, 2, 3, 4, 5, 6...
a3=-2a1
a4=0
a5=3a1
a6=0
a7=-4a1
a8=0
The answer in the book says y=a0sigma from m=0 to infinity of (-1)^m * (2m+1)x2m+a1sigma from m=0 to infinity of (-1)^m * (m+1) x^(2m+1). How do I get the correct answer?

2. Jul 28, 2016

Stephen Tashi

You didn't explain how you got those results. One way to solve this kind of problem is to do manipulations with summations using the $\Sigma$ notation. Another way is to work out cases for specific powers of $x$ until you see how to express the result as a pattern using the $\Sigma$ notation. Which method do your class materials expect you to use ?

Assume $y = A_0 + A_1 x + A_2 x^2 + A_3 x^3 + .... = \Sigma_{i=0}^\infty A_i x^i$. To start things off, what is the summation that represents $y'$ ?

3. Jul 30, 2016

Math10

Just like you mentioned, I got those results by finding y, y' and y'' and then plugging those into the differential equation and simplify. But at the end after finding all the values of a3, a4, a5, etc. I don't know how to get to the correct answer in the textbook.

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4. Jul 30, 2016

Stephen Tashi

The picture of your work is upside down.

5. Jul 31, 2016

Math10

Now I believe you can take a look at my work. Sorry about that.

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6. Aug 1, 2016

Stephen Tashi

After the step:

$\sum_{n=0}^\infty {(n+2)(n+1)A_{n+2}x^n} + \sum_{n=2}^\infty {(n)(n-1)A_nx^n}+ 6 \sum_{n=1}^\infty {(n)A_n x^n }+ 6 \sum_{n=0}^\infty A_n x^n = 0$

I don't see how you concluded $A_0 = 0$

$(2)(1) A_2 x^0 + (3)(2)A_3x^1 + \sum_{n=2}^\infty {(n+2)(n+1)A_{n+2}x^n}$
$+ \sum_{n=2}^\infty {(n)(n-1)A_nx^n}$
$+ (6)(1)A_1x^1 + 6 \sum_{n=2}^\infty {(n)A_n x^n }$
$+ 6A_0 x^0 + 6A_1 x^1 + 6 \sum_{n=2}^\infty A_n x^n = 0$

$( 2 A_2 + 6A_0)x^0$
$+ (6A_3 + 12A_1)x^1$
$+ \sum_{n=2}^{\infty} {[ (n+2)(n+1)A_{n+2} + (n)(n-1)A_n + 6nA_n + 6A_n] x^n} = 0$

$( 2 A_2 + 6A_0)x^0$
$+ (6A_3 + 12A_1)x^1$
$+ \sum_{n=2}^{\infty} {[ n+2)(n+1)A_{n+2} + (n)(n-1)A_n + 6nA_n + 6A_n] x^n} = 0$

$( 2 A_2 + 6A_0)$
$+ (6A_3 + 12A_1)x^1$
$+ \sum_{n=2}^{\infty} {[ n+2)(n+1)A_{n+2} + ( (n)(n-1) + 6n + 6)A_n] x^n} = 0$

7. Aug 2, 2016

Math10

Okay, so from where you left off above,
2a2+6a0=0
that means a2=0 and so does a0=0.
And (6a3+12a1)x=0,
that means a3=-2a1.
Now what? How to find the answer from here?

8. Aug 2, 2016

Staff: Mentor

Look at what you have so far:

$$y = a_0(1-3x^2...)+a_1(x-2x^3...)$$

9. Aug 2, 2016

Math10

Okay, so I got y=a0(1-3x^2-5/3x^4+7/3x^6-3x^8)+a1(x-2x^3+3x^5-4x^7).

10. Aug 2, 2016

Staff: Mentor

The $a_0$ sum does not match the book answer, but the $a_1$ sum does.

11. Aug 2, 2016

Math10

But how do I get to the answer in the book?

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12. Aug 2, 2016

Staff: Mentor

Does the answer in the book satisfy the differential equation?

13. Aug 2, 2016

Math10

To be honest, I'm not too sure...

14. Aug 2, 2016

Staff: Mentor

What do you get if you substitute their answer term for term?

15. Aug 2, 2016

Math10

I still don't get it.

16. Aug 2, 2016

Staff: Mentor

Your methodology looks correct. That's the important part. If their answer is correct and you don't match it, you must have made a simple "arithmetic" error somewhere.

17. Aug 2, 2016

Math10

After checking my work, I think I have made a stupid mistake.
Now I got y=a0(1-3x^2-5x^4+7x^6-9x^8)+a1(x-2x^3+3x^5-4x^7). Does this match the answer in the book?

18. Aug 2, 2016

Stephen Tashi

No, it doesn't imply a definite value for $A_0$ or $A_2$. (After all, the solution to a differential equation should contain some arbitrary constants that are determined only when you are given some initial conditions.)

19. Aug 2, 2016

Stephen Tashi

$A_2 = - 3 A_0$

$A_3 = - 2 A_1$

$A_{n+2} = -\frac{n^2 + 5n + 6}{(n+2)(n+1)} A_n = -\frac{(n+2)(n+3)}{(n+2)(n+1) } A_n$

So we can take $A_0$ and $A_1$ as the arbitrary constants and express all the other $A_i$ in terms of $A_0$ and $A_1$.

20. Aug 3, 2016

Staff: Mentor

You should be able to determine this yourself. The answer does match.