Question

4. Find two linearly independent series solutions of y" + x²y' + xy = 0 (description of the series up to degree 6 is enough.)

Answer 1

0 y" + x²y1 tayo O Let the Solection of the The equation @ is differentiable form yen Ź cm am m=o m-2 ma m=2 Then, y's") = { cm mamt, y" (x) = { m m (mt) * using it in equation 0, we have E m. m (m-1) x M-2 tik Con.mame +x (miem B- = 0 m-o m=1 m=2

mel m(m-1) (mam-2te Cmimames te cm x m1 mno m=2 t zuk (rulum E Am (m+lJames m:2 mo 4 m my mt2 replave m-1 by E m-312 mag (mt) (m +2-1) X" + Cm- mm=0 E=m m=o E Cmta (m+1)(m+a). am te melm-lgumzo a-lu Om w O Cmt 2(muti) (m+2) + Com m) xm mo re Cmta m Cm M+l) (m+2) Equation () is known as recurrence relation. m=l, C₃ Co = - §=-Co (1+1)(1+2) 9 m-2, CA= 2 C .-=ty 3x4 ' {=u ی۔ -3 ¢ 7 C5 = 3 20 G 4 X 5 m24, - 42 CG = ca -2 7 5 = . 5x63 m=5, E to 5 q G= 6x7 - 24 42 'g-u Co -63 ç Co. 7X84 3 २४ S

- Again, Co= -2x - 2 -C 63 17 G = cols Co 45. Cza - 1 /2 x 1/64 G 5 252 G Co 3 28 X G = C = 9 560 20 & Then from o G+C, xt Gx² + S x² + ax + Gust can't Gul + Con 8 + y= Co + Gx tq x² - Cox² + 5 c, x7 *9*'-124*57464 6 + 252 + 500 g ud x3 co (in yen) = + I 95 x .--) + + C 4(2- *** + Xt 252 + & (x2x5 + 8 9 560 & ) + which is required solution for two indefendered Solution choose random. Strandamay as you with wish . #