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Linear Differential Equations of First Order

Solve: xyd y...

Question

Solve:
$x y \frac{d y}{d x} = (1 + x) (1 + y)$

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Solution

$x y \frac{d y}{d x} = (1 + x) (1 + y)$

$\Rightarrow \frac{y d y}{1 + y} = (\frac{1 + x}{x}) d x$
Integrating both sides

$\int \frac{y d y}{1 + y} = \int (\frac{1}{x} + 1) d x$

Let $1 + y = z \Rightarrow d z = d y$

$\Rightarrow \int \frac{(z - 1)}{z} d z = log x + x + C$
$z - log z = log x + x + C$

$1 + y - log (1 + y) = log x + x + C$

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