Question

Find at least the first four nonzero terms in a power series expansion about x = 0 for a general solution to the given differential equation:
y'-e^x y = 0 ; y(0) = 1

Answer 1

\[y'-e^x y= 0 ; y(0)=1\]
@zepdrix

Answer 2

Grr I've kind of forgotten how to do these.
Let's see... so we want our solution to be of the form:\[\Large\rm y(x)=\sum_{n=0}^{\infty} a_n x^n\]Yes?

Answer 3

yes, we can either do it explicitly or find the recursive

Answer 4

So in order to plug stuff in, we also need our derivative.\[\Large\rm y'(x)=\sum_{n=1}^{\infty} n a_n x^{n-1}\]

Answer 5

explicitly? :o

Answer 6

i was kidding about that

Answer 7

So plugging stuff in,\[\Large\rm \sum_{n=1}^{\infty} n a_n x^{n-1}-e^x\sum_{n=0}^{\infty} a_n x^n=0\]Hmm I can't remember this process >.<
Do we want the power series expansion of e^x?

Answer 8

yes I guess so

Answer 9

the prof hasn't taught us yet, but he wants us to fend for ourselves with this problem.

Answer 10

i am a babe in the woods right now.

Answer 11

Example 4 shows it worked out with e^x in the mix.
I'm trying to see if I can make sense of it.

Answer 12

the book jumps around a lot (:((( )

Answer 13

Oh I thought we'd be doing some type of fancy summation product.
But it looks like they're just grabbing the first 4 terms of each expansion and working with those.

Answer 14

So expanding out e^x as powers of x gives us,
\[\Large\rm \sum_{n=1}^{\infty} n a_n x^{n-1}-\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+...\right) \sum_{n=0}^{\infty} a_n x^n=0\]

Answer 15

And then we would distribute the summation to the first few terms of our e^x expansion.

Answer 16

So like for the 1 times summation we would get,\[\Large\rm a_o+a_1 x+a_2 x^2+a_3 x^3+...\]For the x times summation we would get,\[\Large\rm a_o x+a_1 x^2+a_2 x^3+a_3 x^4+...\]For the x^2/2 times summation we would get,\[\Large\rm a_o \frac{x^2}{2}+a_1 \frac{x^3}{2}+a_2 \frac{x^4}{2}+a_3 \frac{x^5}{2}+...\]

Answer 17

Do that up to the 4th degree term and also expand out the derivative summation as well.
Then group like terms together and find your recurrence relation.

It's going to be long and tedious :O
It prolly won't be as bad on paper.
Typing it all out would be a nightmare though lol