Solve x^2y"-4xy'+6y=0 to obtain y=

kindly assist me with the full answer and some explanation.

Thanks in advance

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- Jan 10th 2009, 03:35 AMKayPeedifferencial equation
Solve x^2y"-4xy'+6y=0 to obtain y=

kindly assist me with the full answer and some explanation.

Thanks in advance - Jan 10th 2009, 04:02 AMHallsofIvy
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. - Jan 10th 2009, 07:32 AMKayPeeThanks
Thanks so much.

I'll try and understand it.