Solve x^2y"-4xy'+6y=0 to obtain y=
kindly assist me with the full answer and some explanation.
Thanks in advance
That is an "equi-potential" or "Euler type" equation: each derivative is multiplied by a power of x, the degree the same as the order of the derivative.
If you make the substitution x= ln t will convert it to a linear differential equation with constant coefficients for y as a function of t.
For simple equation, you can try a "trial" solution of the form for some constant r.
If , then and . Putting those into this equation,
= .
In order for that to be 0 for all x, we must have the "characteristic equation", so r= 3 and r= 2.
Both and are solutions to this equation. Since it is a linear homogeneous equation, the general solution is where C and D can be any constants.