Using the power series methods solve the 1st order Lane-Emden Equation: xy = 2y + xy = 0
You may only use a power series solution to find both linearly independent functions. This means you may not use Abel’s theorem, variation of parameters or reduction of order.

Step-by-step explanation:

xy = 2y + xy = 0

Hence, 2y + xy = 0 (1)

Differentiating equation (1) n times by Leibnitz theorem, gives:

2y(n) + xy(n) + ny(n - 1) = 0

Let x = 0: 2y(n) + ny(n - 1) = 0

2y(n) = -ny(n - 1)

∴ y(n) = -ny(n - 1)/2 for n ≥ 1

For n = 1: y = 0

For n = 2: y(1) = -y

For n = 3: -3y(2)/2

For n = 4: -2y(3)

answer: city needs 2.4 inches of rainfall more to match the record of 40 inches.

step-by-step explanation:

since we have given that

total annual rainfall a city record = 40 inches

this year, total rainfall recorded = 37.6 inches

number of inches of rainfall   is needed to match the record of 40 inches is given by

hence, city needs 2.4 inches of rainfall more to match the record of 40 inches.

175

step-by-step explanation:

2 inches x 10 miles = 200 miles - 3 inches = 175