Find the recurrence relation for the coefficients of the power series solution o

Question

Find the recurrence relation for the coefficients of the power series solution of the differential equation y" + y' - 2y = 0 Assume the form y(x) = Sigma^infinity _n = 0 c_nx^n. First compute y'(x) = Sigma^infinity _n = 1 c_nx^n-1 = Sigma^infinity _n = 0 c_n + 1^x^n Then compute y"(x) = Sigma^infinity _n = 1 c_n + 1 x^n - 1 = Sigma^infinity _n = 0 c_n _ 2x^n Then y" + y' - 2u = Sigma^infinity _n = 0 [c_n + 2 + c_n + 1 + c_n]x^n Requiring that the terms of this series for y" + y' - 2y vanish gives the recurrence relation c_n = 2 = c_n + 1 + c_n for n = 0, 1, 2, ...

Explanation / Answer

Given yII+yI-2y=0. ---------> 1

Let y(x)= c0+c1x+c2x2+.....+cnxn+.... be the solution of equation 1

yI(x)=c1+2c2x+..........+(n)cnxn-1+...

yII(x)=2c2+.....+n(n-1)cnxn-2+.....

Substituting these in equation1 and equating the various powers of X, we get

==> [2c2+....+n(n-1)cnxn-2+....] + [c1+2c2x+......+ncnxn-1+..... ] -2[c0+c1x+c2x2+......+cnxn+.....]=0

2c2+c1-2c0=0

==> c2=(-1/2) c1+ c0

substituting these in y(x) we get

y(x)=c0+c1x+(-1/2c1+c0)x2+.....

   =c0(1+ x2+.....)+c1(x-(1/2)x2+...)

fill up blanks

(i) n ,n+1

(ii) n(n+1), (n+1)(n+2)

(iii) cn+2=(1/(n+1)(n+2))[-(n+1)cn+1-ncn)

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