suppose that (a-1)^2=4B. Using reduction of order, show that (ln x)x^((1-a)/2) is a 2nd solution of Euler's

Question

suppose that (a-1)^2=4B. Using reduction of order, show that (ln x)x^((1-a)/2) is a 2nd solution of Euler's equation x^2y''+axy'+By=0

Answer 1

y1(x) = x^((1-a)/2) is the first solution, which can be obtained by assuming a solution of the form x^c exists and solving for c. Assume the second solution has the form

y(x) = v(x)y1(x)

and substitute this into the equation. You have

y' = y1'v + vy1'
y'' = y1'' v + 2v'y1' + y1v''

so

0 = x^2(y1''v + 2v'y1' + y1v'') + ax(y1'v + v'y1) + Bvy1
= [v(x^2 y1'' + axy1' + By1)] + x^2(2v'y1' + y1v'') + axv'y1
= x^2(2v'y1' + y1v'') + axv'y1
= x^2y1v'' + (2x^2y1'+axy1)v'

Now, let w = v'. You then have a first-order equation in w,

x^((5-a)/2)w' + x^((3-a)/2)w = 0,

after substituting y1 = x^((1-a)/2) and simplifying.

Divide through by x^((5-a)/2) and rearrange:

w' = (-1/x)w.

Then w = e^(-ln x)w0 = w0/x, for some constant w0. Therefore v = w0*ln x + C, where C is another arbitrary constant. Setting w0=1 and C=0 gives the solution you seek.