(1) show that there is a regular singular point at x0 = 0 and determine the interval on which any Frobenius series solution must hold (2) find two linearly independent Frobenius series solutions if possible, if not, find one and show why you can not get a second one. Be sure to give the recursion formulas and exponents x2y\" + xy\' + (x4 - 4)y = 0 Solution Divide by x square the given equation to get y\" +y\'/x +(x^2-4/x^2) y =0 As x is in denominator, x=0 makes the equation undefined. Hence regular singular point at x =0 So we split the domain as x>0 and x <0.

1. (1) The following table summarizes information which are associated with three new 3D Printers
being considered for use in a manufacturing plant. Note that M&O stands for Maintenance &
Operation Cost.
The company's interest rate (MARR) is 12%. Which 3D Printer should the company choose?
Use Annual Cash Flow Analysis.
Solution
NOTES:
(1) M&O is an annual cost.
(2) M&O Gradient: The M&O gradient is 0 at the end of year 1 & increase by the value of
gradient every year. For example, for Printer 1, M&O = $78,000 for year-1. Then, M&O for
year-2 = $78,000 + $15,000 = $93,000. M&O for year-3 = $93,000 + $15,000 = $108,000, and
so on.
(3) Depreciation being a non-cash expense, would be excluded from the annual cash flow
considerations.
Company will choose that printer for which the NPV (Sum of present values of all cash inflows
& outflows) is highest.
The calculations & results are as follows.
From above results, we see that NPV of the 1st Printer is highest, at $619,248. So the company
will choose printer 1.Printer 1YearInvestmentAnnual BenefitAnnual M&OSalvage ValueNet
Annual Cash InflowDiscount Factor @12%Discounted Cash Flow (Present Value)0-27,80,000-
27,80,0001.0000-
27,80,00016,70,00078,0005,92,0000.89295,28,57126,70,00093,0005,77,0000.79724,59,98136,7
0,0001,08,0005,62,0000.71184,00,02046,70,0001,23,0005,47,0000.63553,47,62856,70,0001,38,
0005,32,0000.56743,01,87166,70,0001,53,0005,17,0000.50662,61,92876,70,0001,68,0005,02,0
000.45232,27,07986,70,0001,83,0004,87,0000.40391,96,69196,70,0001,98,0004,72,0000.36061
,70,208106,70,0002,13,0004,57,0000.32201,47,142116,70,0002,28,0004,42,0000.28751,27,064
126,70,0002,43,0004,27,0000.25671,09,600136,70,0002,58,0004,12,0000.229294,420131,18,00
01,18,0000.229227,043NPV6,19,248Printer 2YearInvestmentAnnual BenefitAnnual
M&OSalvage ValueNet Annual Cash InflowDiscount Factor @12%Discounted Cash Flow
(Present Value)0-26,30,000-26,30,0001.0000-
26,30,00016,50,00071,0005,79,0000.89295,16,96426,50,00083,5005,66,5000.79724,51,61036,5
0,00096,0005,54,0000.71183,94,32646,50,0001,08,5005,41,5000.63553,44,13356,50,0001,21,0
005,29,0000.56743,00,16966,50,0001,33,5005,16,5000.50662,61,67576,50,0001,46,0005,04,00

2. 00.45232,27,98486,50,0001,58,5004,91,5000.40391,98,50996,50,0001,71,0004,79,0000.36061,
72,732106,50,0001,83,5004,66,5000.32201,50,201116,50,0001,96,0004,54,0000.28751,30,5141
197,00097,0000.287527,885NPV5,46,702Printer 3YearInvestmentAnnual BenefitAnnual
M&OSalvage ValueNet Annual Cash InflowDiscount Factor @12%Discounted Cash Flow
(Present Value)0-23,00,000-23,00,0001.0000-
23,00,00015,80,00065,0005,15,0000.89294,59,82125,80,00076,0005,04,0000.79724,01,78635,8
0,00087,0004,93,0000.71183,50,90845,80,00098,0004,82,0000.63553,06,32055,80,0001,09,000
4,71,0000.56742,67,25865,80,0001,20,0004,60,0000.50662,33,05075,80,0001,31,0004,49,0000.
45232,03,10585,80,0001,42,0004,38,0000.40391,76,90195,80,0001,53,0004,27,0000.36061,53,
980982,00082,0000.360629,570NPV2,82,699