(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] !!!Power Series Solution to a Diff EQ!!!

1. The problem statement, all variables and given/known data

Find the first 5 term of a Power series solution of

[tex]y'+2xy=0[/tex] (1)

Missed this class, so please bear with my attempt here.

3. The attempt at a solution

Assuming that y takes the form

[tex]y=\sum_{n=0}^{\infty}c_nx^n[/tex]

Then (1) can be written:

[tex]\sum_{n=1}^{\infty}nc_nx^{n-1}+2x\sum_{n=0}^{\infty}c_nx^n=0[/tex]

Re-written 'in phase' and with the same indices (in terms of k):

[tex]c_1+\sum_{k=1}^{\infty}(k+1)c_{k+1}x^k+\sum_{k=1}^{\infty}2c_{k-1}x^k=0[/tex]

[tex]\Rightarrow c_1+\sum_{k=1}^{\infty}[(k+1)c_{k+1}+2c_{k-1}]x^k=0[/tex]

Now invoking the identity property, I can say that all coefficients of powers of x are equal to zero (including [itex]c_1*x^0[/itex])

So I can write:

[itex]c_1=0[/itex] and

[tex]c_{k+1}=-\frac{2c_{k-1}}{k+1}[/tex]

Now I am stuck (I know I am almost there though!)

Should I just start plugging in numbers for k=1,2,3,4,5 ? Will this generate enough 'recursiveness' to solve for the 1st five terms?

Is that the correct approach?

Thanks!!

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# Homework Help: Power Series Solution to a Diff EQ

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