Question

Q1 Develop the series solutions of the following differential equations by Frobenius Method (1) **"+2 y' + x y = 0 xy" + y = 0 ry" - (2x - 1) y' + (x+1)y = 0 xy"+y' - x y = 0 2x (x - 1)y" -(x+1) y' + y = 0 xy" +2 y' +16 x y = 0 (iv) (v) (vi)

Answer 1

| Page No. so ntry 8 n=o nto-2 x = intr-1) Intro y antr- gines the indreal conation Given xyl +2y + xy=0 is regular singular poshut of (1) the Equation admits of Frobenius Sexies of the form y = ² con anto cozo which converges for all x from (2) a entr) en x yl - Ž (n+8-1)(n+r) en x in (1) Using these obtion +2 z (n+8) +* I Can yht z (nto) (n+8-1) en à lowest powers equation so coffrerent we moo ntr- x 1O + 2 Cn7741 15

Page Noi 8=0,-) nave from (3) we recursive formula. following Intr + 1) (n+8+2) Entl teng 1 Enti Cn-l (n+&+1) (n+8+2) we have from (4) C = C₂ = 5 (8+2)(8+3) Cy = (8+4)(875) (x+2)(x+3) (x+y)(x+5) 1 (7+2)(773) -1 10+2)(+3) (0+y)(4+5)

Page No: Now for x2 x4 + 120 21 2 x4 = 4, (n) = Co 1 - + 31 5 Since other is an integer maependent soution is . ray orx Jy zo => 1 y(x) = y, lx) login) 2 5 36 . 03 General Solution yen) = Ay, (x) + B Y (7) A, B are constants Thank you!

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