You should get : .
so that the two solutions are : & .
Hence, the general solution is .
ok i wanted to know if someone could check my work and let me know if they got the same thing.
im supposed to find the values of m so that the function y = x^m is a solution to,
x^2y'' - 3xy' - y = 0.
i got m = 0. i just took the first and 2nd derivative of the function and plugged it into the equation and solved.
thanks in advance.
Generalization :
In case we don't know there are solutions of the form and considering that we have an Euler equation, the subtitution with independent variable, transforms the given equation into a linear homogeneous differential equation with constant coefficients.
Fernando Revilla