xy'' - y = 0?

Question

Find the series solution of the differential equation.

Answer 1

Let y = sum k=0 to infinity ak x^k. Then
y'' = sum k=2 to infinity ak*k*(k-1)x^(k-2), and the equation becomes

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

The first sum can be rewritten as
sum k=1 to inf ak+1*(k+1)*k*x^k,
so by combining like terms, you get

a0 = 0, ak+1*(k+1)*k = ak for k >= 1, and a1 is arbitrary.

Therefore, the solution is
a1 * sum k=1 to infinity x^k / (k!(k-1)!)

Answer 2

what r u solving for?