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### Messages - Changyu Li

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16
##### Ch 1--2 / Re: Bonus problem for week 2
« on: January 18, 2013, 01:41:07 AM »
$x = u+h \\ y = v+k\\$
at $(u,v) = 0\\$
$k - h - 2 = 0 \\ h + k = 0 \\ \Rightarrow h = -1,\;k = 1\\$
$x = u - 1 \\ dx = du\\ y = v + 1 \\ dy = dv \\$
$$\frac{dv}{du} = \frac{v-u}{u+v} \\$$

let $v = ut,\;\frac{dv}{du}=t+u \frac{dt}{du}$

$$t + u \frac{dt}{du}=\frac{ut-u}{u+ut} = \frac{t-1}{1+t}$$
simplify with magic
$$\frac{1}{u} du = \frac{1+t}{-1-t^2}dt \\ \ln \left| u \right| = -\frac{1}{2}\ln \left| t^2 +1 \right| -\arctan t + C \\ \ln \left| u \right| = -\frac{1}{2}\ln \left| \left( \frac{v}{u} \right) ^2 +1 \right| -\arctan \left( \frac{v}{u} \right) + C \\ \ln \left| x+1 \right| = -\frac{1}{2}\ln \left| \left( \frac{y-1}{x+1} \right) ^2 +1 \right| -\arctan \left( \frac{y-1}{x+1} \right) + C$$

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