Please see the attached file for the fully formatted problems.
Find a solution in the form of a power series for the equation
y" - 2*x*y' = 0
(ie find 2 linearly independent solutions y1(x) and y2(x)).
After doing that, note that the equation can also be solved directly by integration:
y"/y' = 2x
ln(y') = x^2 + c1
y' = ke^(x^2) k=e^c1
y = k* integral((e^(t^2))dt + C) from 0 to x
Thus, one of your power series solutions gives and explicit form for the integral:
integral(e^(t^2)dt) from 0 to x
A solution is found in the form of a power series for an ODE.