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## Homework Statement

Find a second solution y2 for x^2*y"+xy'-y=0; y1=x that isn't a constant multiple of the solution y1.

## Homework Equations

None.

## The Attempt at a Solution

Here's my work:

I divided by x^2,

y"+(1/x)y'-(1/x^2)y=0

P(x)=1/x and Q(x)=-1/x^2

Let y(x)=v(x)*x

y'(x)=v'(x)*x+v(x)

(1/x)y'(x)=v'(x)+v(x)/x

y"(x)=v'(x)+v"(x)*x+v'(x)=xv"(x)+2v'(x)

xv"(x)+2v'(x)+v'(x)+v(x)/x-v(x)*x/(x^2)=0

xv"(x)+3v'(x)=0

Let w=v'

w'=v"

xw'+3w=0

w'=-3w/x

dw/dx=-3w/x

dw/w=-3/x dx

integrate

ln abs(w)=-3ln abs(x)+C

Don't count C, the constant.

w=1/x^3

w=v'=1/x^3

dv/dx=1/x^2

don't count c, the constant.

v=-1/2x^2

y=v*x

y=-1/2x^2*x=-1/2x

But the answer in the book is y2=1/x. I got y=-1/2x, which answer is right?