xy'' + (1-x)y' + 4y = 0?

Question

Show that the differntial equation has a polynomial solution (i.e., the infinite series terminates after finitely many terms).

Answer 1

Let y = sum k=0 to inf ak*x^k.

Substitute this into the equation and you get

sum k=2 to inf ak*k(k-1)x^(k-1) +
sum k=1 to inf ak*k*x^(k-1) -
sum k=1 to inf ak*k*x^k +
sum k=0 to inf 4ak*x^k = 0.

Rewrite the sums by shifting indices:

sum k=1 to inf ak+1*k(k+1)x^k +
sum k=0 to inf ak+1*(k+1)*x^k -
sum k=1 to inf ak*k*x^k +
sum k=0 to inf 4ak*x^k = 0.

Therefore, a1 = -4a0, and
ak+1*(k+1)^2 + (4-k)*ak = 0. That is,
ak+1 = (k-4)*ak / (k+1)^2, but then the k-4 factor causes a5 = 0, and ak+1 = constant * ak, so all subsequent terms will be zero.

Answer 2

yea u shud probly study harder and u also need to see what variable or whatever ur solving for or else u can only work it out as far as
-4xy+1=0

Answer 3

This is like, the third question that you have asked for someone to do your homework for you. You will never learn that way.