Consider the differential equation y" − 2xy' + y = 0, y(0) = 0, y' (0) = 1. Assume a power series solution of the form

Question

The Form

∞

y =∑ a_nxⁿ

n=0

Find the values of a₅ and a₇.

Answer 1

Since you only need a₅ & a₇, take y'' = 2xy - y we know that y(0) = 0 & y'(0) = 1

The solution is y = f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f^iv(0)x⁴/4! + f^v(0)x⁵/5! + f^vi(0)x⁶/6! + f^vii(0)x⁷/7!......

with a₅ = f^v(0)/5! and a₇ = f^vii(0)/7!

we know that y''(0) = f(0) = 2(0)(0) - (0) = 0 . Taking the other derivatives in similar fashion and finding a₅ & a₇, we find that

a₅ = 5/5! = 1/4! = 1/24 and

a₇ = 45/7! = 1/112