Given that a is a constant, use power series to find a solution of the following initial value problem.

Question

Give first 4 non-zero terms:

y'' + xy' + y = 0 y(0) = a y'(0) = 0

Answer 1

The easiest way to establish a power series is to formulate a Taylor series. Since your initial conditions depend on x = 0, the series will be centered at this point. The series will be of the form

y = f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f^iv(0)x⁴/4!.................

the first term is known, f(0) = a write the equation as y'' = -xy' - y

So,

y''(0) = -(0)(0) - (a) = - a

Then,

y''' = -y' - xy'' - y' then y'''(0) = -(0) - (0)(-a) - (0) = 0

Then,

y^iv = -y'' -y'' - xy''' - y'' then y^iv(0) = +2a - (0)(0) + a = 3a

Then,

y^v = -4y''' - xy^iv then yv(0) = -4(0) -(0)(-a) = 0

Then,

y^vi = -5y^iv - xy^v then y^vi = -5(3a) - (0)(0) = -15a

So, the first 4 terns would be

y = f(x) = a + (-a)x²/2! + (3a)x⁴/4! + (-15a)x⁶/6!