i need with part (c). i have given my answers to part a) and b) for context.
consider the euler-cauchy differential equation x^2y'' + axy' + by = 0 where x< 0.
(a) i have used the ansatz y = (-x)^r to get the characteristic equation r^2 + r(a-1) + b =0.
(b) i have done transformation via transformation of variables with x=-e^t. this resulted the a second order homogeneous d. e. : d2y/dt2 + (a+1)dy/dt + by = 0.
(c) determine the characteristic equation for the equation found in part (b). compare with part (a).
are they the same? how do i get the characteristic equation from the differential equation i got in part (b)?