Thread: some trouble with this differencial equation

ok i really hate asking these series question because i know how much of a pain they are to answer on this thing. ok this questions requires the use of the frobenius method to find the 2 independent solution of the following equation.

(X^2)y"+(e^-x)y'-y=0 around the point x=0

well if someone could give me a hand with this one it'll me nice, if not can someone else tell me the series of e^-x is because i do not recall it because you need to write e^-x as a series that i know.

i got one more,

3(x+2)^2y" + 4(x+2)y' + xy = 0

this one also asks for 2 independent solution but i don't think you need the forbenius method for this, if some one could just give be some quick direction one how to solve this one i don't need a full solution. Thanks alot

Originally Posted by action259

ok i really hate asking these series question because i know how much of a pain they are to answer on this thing. ok this questions requires the use of the frobenius method to find the 2 independent solution of the following equation.

(X^2)y"+(e^-x)y'-y=0 around the point x=0

well if someone could give me a hand with this one it'll me nice, if not can someone else tell me the series of e^-x is because i do not recall it because you need to write e^-x as a series that i know.

i got one more,

3(x+2)^2y" + 4(x+2)y' + xy = 0

this one also asks for 2 independent solution but i don't think you need the forbenius method for this, if some one could just give be some quick direction one how to solve this one i don't need a full solution. Thanks alot

Define S(f(n),n,a,b) to be the sum of f(n) as n varies from a to b.

The Frobenius method for an expansion about x = 0 says the solution of the DEq will be:
y(x) = S(a_n * x^{k+n},n,0,infinity) = a_0 * x^k + a_1 * x^{k+1} + ...
where the a_n are constants, and k is a constant (I believe it is an integer, but I might be wrong about that.)

What you need to do is find y' and y'' and then manipulate the sums so that you have only one summation left over. If you can stomach it, I already did an example of this here.

Also:
e^{-x} = S([-1]^{n}*x^n /n!,n,0,infinity) = 1 - x + (1/2)x^2 - (1/6)x^3 + ...

If you have any questions about this or the example link, just let me know.

-Dan