Your title says "first six coefficients". Coefficients of what? I suspect you were expected to find a power series sollution.
There are two ways you can do that:
1) Write the solution as a power series: so the and .
Put those into the differential equation to get a "recursive equation" for the coefficents . From y(0), = 0. From y'(0)= 1, [tex]a_1= 1[tex]. Then determine , , and .
2) Recall that the first 6 terms of a McLaurin series for y is . Then , . For higher derivatives, evaluated at x= 0, solve the equation for y'': y''= (-xy+ y')/(1- x) so y''(0)= (y')/1= 1,
so that , etc.