1. The problem statement, all variables and given/known data The function satisfies the differential equation f''(x) = xf(x) and has boundary conditions f(0) = 1 and f'(0) = 1 Use Frobenius method to solve for f(x) with a taylor expansion of f(x) up to the quartic term a4x4 2. Relevant equations f(x) = a0 + a1x + a2x2 + a3x3 + a4x4 f'(x) = a1 + 2a2x + 3a3x2 + 4a4x3 f'(x) = 2a2 + 6a3x + 12a4x2 3. The attempt at a solution What I have done so far is set the differential equation to zero and applied the series to f''(x) and xf(x) which gave an series of a0x + a1x2 + a2(x3 +2) + a3(x4 + 6x) + a4(x5 + 12x2) = 0. I then set up equations for each power of x which is equal to zero. From the x powers I get a2 = 0, a3 = -a0/6, and a4 = -a4/12. Applying the boundary conditions shows that a0 = 1 and a1 = 1. When I use the values in the equation, the fractions within each power of x cancels each other out and you are left with no series and just a trivial statement of 0 = 0. What am I doing wrong in trying to find a series that describes the function produced by the differential equation?