Attached are two problems, one with an answer that I don't understand how it was derived and one problem without the answer that I would like to see how it is solved.
PowerSeries Methods - Introduction and Review of PowerSeries
14. Find two linearly independent powerseriessolutions of the given differential equation.

Consider the differnetial equation
y'(x) + xy(x) = 0 with y(0) = 0
Look for a solution of this problem of the form
y(x) = A + B + Ce^-x + De^-1/2x^2
Use the fact that y must satisfy the equation andtheinitial conditions to identify the constants A,B,C and D. By setting u = -x^2/2 in thepowerseries for f(u) = exp{u},

Please see the attached file for the fully formatted problems.
Find a solution in the form of a powerseries 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 +

By considering appropriate series expansions, prove that thepowerseries expansion of the product of the (infinitely many) exponential factors e^{(x^i)/i}, i = 1, 2, 3, ..., is 1 + x + x^2 ... for |x| <1.
By expanding each individual exponential factor in the product and multiplying out, also show that the coefficient of x^1

Consider the diffusion equation
ut = ku.xx for 0 < < pi and t > 0 with the boundary conditions
ux(0, t) = 0 and u(pi, t) = 0
andtheinitial condition
u(x,0) = 1.
(a) Find the separated solutions satisfying the differential equation and boundary conditions.
(b) Use these solutions to write an explicit series solution to t

Hello,
I am in a fast-paced Calculus course where I must learn new concepts each week; I find it challenging to grasp the concepts while remaining on-pace and I am experiencing great difficulty. I have a few weeks before my semester is over, and thankfully, I have a passing grade.
I am really finding it difficult to grasp