- #1

Jtechguy21

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## Homework Statement

Dy/Dx = (Y-x)/(Y+x)

## Homework Equations

Y=ux

dy=udx+xdu

## The Attempt at a Solution

Dy/Dx = (Y-x)/(Y+x)

Plug in my substitutions

udx+xdu(1/dx)=(ux/ux+x) - X/(ux+x)

Simplify

u+x(du/dx)=(ux)/x(u+1) - (x)/((x)(u+1))

u+x(du/dx)=u/(u+1) -(1)/(u+1))

u+x(du/dx)=u-1/(u+1)

This is where I think i begin to mess up

u+du=(u-1)/(u+1) dx/x

substract (u-1)/(u+1) to the other side

u-(u-1)/(u+1) du=dx/x

I know the right side integrates to Lnx +c

but on the left side if i do

(u^2-1)/(u+1)

I split it up into

the integral (u^2)/(u+1) minus integral of 1/(u+1)

(u^2)/(u+1)<-use long division

I get u+(1/u+1) minus the integral of 1/(u+1)

i am left with just the integral

of u

u^2/2= lnx+c

plug u back in.

((y/x)^2)/2 =lnx +c

is this sufficient of an answer?

according to the answer key I am going to end up with the arctan somewhere in my answer. so i may have already messed up :(

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