Question

Solve near point 0
x^2y" + y =0
I got characteristic equation r^2-2+1 =0 and 2 complex roots are r = \(\frac{1}{2}\pm \frac{\sqrt{3}}{2}i\) but I don't know how to construct the series. Please, help

Answer 1

Here what I went so far
check regular singular point:
\[lim_{x\rightarrow 0}x^2\frac{1}{x^2}=1 \] so, x =0 is regular singular point.
and the stuff above.

Answer 2

This is an Euler's differential equation. You can solve using http://en.wikipedia.org/wiki/Euler%27s_differential_equation . Solution will involve x^r

Answer 3

You can probably use the limit idea or linearize around zero using taylor expansion

Answer 4

thanks a lot. I will.

Answer 5

You shouldn't be getting a characteristic equation here...

Answer 6

why? that 's what I read from the book

Answer 7

is it Euler's form?

Answer 8

No, the characteristic equation (only) arises from an equation that is linear, homogeneous, and has constant coefficients.

Answer 9

if you said so, what should I do? construct as y = sum a_n x^n?

Answer 10

That's what I would do, given that the instruction say "Solve near point 0." Otherwise, @ybarrap is right about the \(y=x^r\).

Answer 11

Cannot be \[y =\sum_{n=o}^{infty} a_nx^n\]because after putting x^2y" +y =0 I get nothing.

Answer 12

Let sumation aside, y = a_n x^n
y'= n a_n x ^(n-1)
y"= n (n-1) a_n x ^(n-2)
and then x^2y"= n(n-1) a_n x^n
what can I get when putting them into the original equation?

Answer 13

Hmm, in that case stick with ybarrap's suggestion.

Answer 14

so? don't have series solution, right?

Answer 15

Yes, it would seem so...

Answer 16

Ok, thanks

Answer 17

@Microrobot

Answer 18

Hi Loser, how can I help?

Answer 19

what do you mean by "how can"? you don't know this stuff?

Answer 20

I have a question: 1.2 E -7 means?

Answer 21

You still working this problem?

Answer 22

.00000012

Answer 23

Take the decimal point 7 units to the left, when minus, to the right if plus

Answer 24

yes, many other thing. and another question. why those are the same? please help me figure out

Answer 25

you mean, scientific notation vs. decimal format?

Answer 26

1) \[38*10^6 e^{-4*10^7 t}-38*10^6e^{-20*10^5t}+12*10^{-8}\]
t = 0.001
and t = 0.01
I got the same answer , ha!! I know it's wrong but why and how to get right?

Answer 27

So with something like this it's very easy to make a calculator mistake. You know how to compute it, just need to go slow sometimes. Let's take this apart.

Answer 28

First term:
$$\Large \bf 38*10^6e^{-4*10^7t}$$
Plug it t=.001 and we'll compare.
Let me know when you are done.

Answer 29

0

Answer 30

how can it be, right? but I did it twice, still get 0

Answer 31

I get for t=.001:
$$\bf \Large
6.31695... × 10^{-17365}$$
What do you get for \( \bf 38*10^6\)?

Answer 32

38 and 6 zero after it.

Answer 33

I think I see the problem.
Your calculator can not handle large numbers. Try google:
https://www.google.com/#bav=on.2,or.r_cp.r_qf.&fp=584e6f03ba966077&q=38*10%5E6

Answer 34

Google's not any better.

Try wolfram:
http://www.wolframalpha.com/input/?i=38*10%5E6e%5E%28-4*10%5E7%28.001%29%29

Answer 35

yes, the same me 1.2 *10^-7

Answer 36

What is this for? The total or just for the 1st term?

Answer 37

I don't know what to do next, can you help me check my work? especially at \(\lambda_ 2\) from my calculator, it showed me -20. 00008. but I really don't know how much it is, I assume that it is -20*10^5 , or it's just -20 ?? shame on me, differential equation student doesn't know how to read a number!!!

Answer 38

For \(\lambda_2\) I get -20.

Answer 39

Ok, thanks, I will redo my work. Ha, it leads to the wrong answer for the rest part, right??

Answer 40

so, lambda 1 is correct, right?

Answer 41

No, just checked with wolfram:
http://www.wolframalpha.com/input/?i=%28-5*10%5E6%2Bsqrt%285*10%5E12-4*10%5E8%29%29%2F2
http://www.wolframalpha.com/input/?i=%28-5*10%5E6%2Bsqrt%285*10%5E12-4*10%5E8%29%29%2F2
lambdas are: -1.38 × 10^6 and -3.62 × 10^6

Answer 42

y(t) sort of cut off, but this is the final solution along with steady state:
$$

\small y(t)=\frac{3 \left(\frac{250}{\sqrt{62499}}-1\right) \left(\left(124999+500 \sqrt{62499}\right) \left(-e^{-\frac{10000 t}{250+\sqrt{62499}}}\right)+e^{-10000
\left(250+\sqrt{62499}\right) t}+500 \sqrt{62499}+124998\right)}{50000000}
\\\\
$$
Steady-state:
$$
\bf
\lim_{t\to\infty}y(t)=\frac{3 \left(\frac{250}{\sqrt{62499}}-1\right)(500 \sqrt{62499}+124998)}{50000000}

$$
Details:
http://www.wolframalpha.com/input/?i=%28-5*10%5E6-sqrt%285*10%5E12-4*10%5E8%29%29%2F2

Answer 43

Im so sorry Loser. I was afk for some time.

Answer 44

I meant by how can I help you is what you need.

Answer 45

Sorry about that :(

Answer 46

@ybarrap Thank you so much for helping him while I was gone!

Answer 47

ur welcome