Problem 08
$xy^2 \, dx + e^x \, dy = 0$, when $x \to \infty$, $y \to \frac{1}{2}$.
Solution 08
$xy^2 \, dx + e^x \, dy = 0$
$\dfrac{xy^2 \, dx}{y^2 e^x} + \dfrac{e^x \, dy}{y^2 e^x} = 0$
$\dfrac{x \, dx}{e^x} + \dfrac{dy}{y^2} = 0$
$\displaystyle \int xe^{-x} \, dx + \int y^{-2} \, dy = 0$
For $\displaystyle \int xe^{-x} \, dx$
Let
$u = x$, $du = dx$
$dv = \int e^{-x} \, dx$, $v = -e^{-x}$