The function y x4 is a solution of the differential equatio

The function y = x^4 is a solution of the differential equation x^2y\'1 - 7xy^1+ 16y = 0 Find the second linear independent solution (using the formula from reduction of order method) and its interval.

Solution

Solution : let x = e^{z ,} xy \' = D y and x²y \'\' = D (D -1) y where D = d/dz and z= log x

The given DE : x² y \'\' - 7x y \' + 16y =0 can be written as

: D ( D - 1) y - 7 D y +16y =0

=> [ D² - 8D +16] y =0

the auxillary equation is m² - 8m + 16 =0 => m = 4, 4 repeated values

hence the solution is : y = ( A + B z ) e^4z = Ae^4z + B ze^4z substituting for z

: y = A x⁴ + B x⁴  logx where A and B are constants , both the parts of the solution

x⁴ , and x⁴ logx will satisfy the DE