Question

x = 0 is an ordinary point of a certain linear differential equation. After the assumed solution y = ∞ n = 0 cnxn is substituted into the DE, the following algebraic system is obtained by equating the coefficients of x0, x1, x2, and x3 to zero.

2c2 + 2c1 + c0 = 0

6c3 + 4c2 + c1 = 0

12c4 + 6c3 + c2 − 1 3 c1 = 0

20c5 + 8c4 + c3 − 2 3 c2 = 0

Bearing in mind that c0 and c1 are arbitrary, write down the first five terms of two power series solutions of the differential equation.

y1(x) = c0 1 − 1 2 x2 + 1 3 x3 − 1 4 x4 + 1 5 x5 + and y2(x) = c1 x − x2 + 1 2 x3 − 1 3 x4 + 1 4 x5 +

y1(x) = c0 1 − x2 + 1 2 x3 − 5 36 x4 − 1 360 x5 + and y2(x) = c1 x − 1 2 x2 + 1 3 x3 − 1 8 x4 + 1 60 x5 +

y1(x) = c0 1 − 1 2 x2 + 1 3 x3 − 1 8 x4 + 1 60 x5 + and y2(x) = c1 x − x2 + 1 2 x3 − 5 36 x4 − 1 360 x5 +

y1(x) = c0 1 − 1 2 x2 + 1 6 x3 − 1 24 x4 + 1 120 x5 + and y2(x) = c1 x − 1 2 x2 + 1 6 x3 − 1 24 x4 + 1 120 x5 +

y1(x) = c0 1 − x2 + 1 2 x3 − 1 3 x4 + 1 8 x5 + and y2(x) = c1 x − x2 + 1 2 x3 − 5 36 x4 − 1 360 x5 +

Answer 1

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Consider the equations obtained in solving power series solution of DE: 2C+2C, +Co = 0 6C, +4C, +C =0 12C, + 6C, C =0 2003 +8C, +C; C =0 Here, Co,C, are arbitrary constants. So, solve remaining terms in terms of Co,C. From the equation 2C2 +2C, +Co = 0, 20, = -2C, -Co @=-6, - 1/6 From the equation 6C, +4C, +C =0, 6C; =-4C, -C, 6C, --- --- -C, = 4C, +2C, -C 6C, = 3C, +2C, C;=<COD om the equation 12C, +C; C = 0, 12C, -6C, 120.=-0 (2-) =-36, -20, +, 120, = _Ç-2C, From the equation 200, +8C, +C, C = 0, 200, = -8C, C 200, ==${*}<-...), %{-C-C.) 200, 406, +6C, C

168 C 20. 2 Cz = - +-- -- 9.20 .6.20 2.20 3.20 3.20° C3 = 40,+ 1c - C 1c 45° 15° 40 30 30° t-Co 36097 360 30 k=0 Therefore, the power series solution of the differential equation is v=3CM = Cost° +Cjx' +Cx? +Cx* +Cr* +Cx* +- =C,·1+C*'+(-4,-3C)x+(4) =C,+C8-Cyr- Corte extra 2 3 1, 14 12 13 2 11 4 14 1 +--- +C x - x + = x - - x + = 360 x +- 9 Er +=x -= x + = x 2 3 6 30 as y =Coy1 (x)+Cyyz (x) Here, ao and ai are arbitrary constants and 1 13 14 - - X + -X -x* + x » (x)=1-6x2 + x - x + box + + ... 6 y, (x) = x-x + 211 x - x + 2 9 360 x +...