Question

use order of reduction formula to find 2nd linearly independent solution to de x^2y" - 9xy' + 25y = 0 if y(x) = x^5 is one solution to the de

Answer 1

Do y Guinen = 0. a one solution By differential equation in 2²y" - gxy' +25y=0 Yox)= 25 is one solution. Reduction formula if y(x) in the second solution y(x) is y (x) = Y, (x) s 4,2x) where equation is of the form y"+ P(x)yl + Q(x) y =0 then, e-Sperode odr. NOL, We have, z?y" – gxy'+25y = 0 y- czy' + 2 y = 0 p(a) = -2 Q(x) = 25 X2 Second solution is -52.de e .da 4 (0) = x 5 S (25)2

sa adx 5 = (2) χ dy 20 gen(x) e dx x 10 20 Toes Jose In x9 e dr . 210 5 ida 1 11 x 2 g z x 10 s tada - y («). = 25 x Y(x) = x5 ln (x) This in required second solution.