Given the information, find a second solution to the differential equation.

Question

anonymous · Answer

>_<

anonymous · Answer

what information was i given?

anonymous · Answer

Gotta give me time to type it up :)

anonymous · Answer

Ohhh, okay

anonymous · Answer

$$xy''+2xy' + 6e^{x}y = 0$$
Roots of this problem are r = 0, 1
The recurrence relation is for r= 1 is:
$$a_{n} = \frac{ -2na_{n-1}-\sum_{k=0}^{n-1}\frac{ 6a_{n-k-1} }{ k! } }{ (n+1)(n) }$$
The first several terms for the solution given this recurrence relation are:
$$y_{1}(x) = x-4x^{2}+\frac{17}{3}x^{3}-\frac{47}{12}x^{4}+...$$
So, using this information to help, I need a second solution of the form:
$$y_{2}(x) = \alpha y_{1}(x)\ln|x| + \sum_{n=0}^{\infty}c_{n}x^{r_{2}+n}$$

anonymous · Answer

@LenaBot4000 Sorry for the time it took to type this.

anonymous · Answer

no problem, seems like it would be a lot to type. And i am sorry for being of no assistance. I need to finish a report in my English class. Would have been glad to help though, farewell and good luck with this

anonymous · Answer

Thank you for looking at it anyway.