# Find the general solution of these first order differential equations (1

Find the general solution of these first order differential equations
$\left(1–x\right){y}^{\prime }={y}^{2}$
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Given that:
$\left(1–x\right){y}^{\prime }={y}^{2}$
Consider
$\left(1-x\right){y}^{\prime }={y}^{2}$
$\left(1-x\right)\frac{dy}{dx}={y}^{2}$
$\frac{dy}{{y}^{2}}=\frac{1}{1-x}dx$
Integrate on both sides
$\int \frac{1}{{y}^{2}}dy=\int \frac{1}{1-x}dx$
Let
$-du=dx$
Hence we have
$\int \frac{1}{{y}^{2}}dy=-\int \frac{1}{u}du$
$\left[\because \int {y}^{n}dn=\frac{{y}^{n+1}}{n+1}+c\right]$
$\frac{{y}^{-2+1}}{-2+1}=-\mathrm{ln}\left(1-x\right)+c$
$-\frac{1}{y}=-\mathrm{ln}\left(1-x\right)+c$
$⇒y=\frac{1}{\mathrm{ln}\left(1-x\right)-c}$
$\therefore$ The general solution is
$y=\frac{1}{\mathrm{ln}\left(1-x\right)-c}$