Question

solve the differential equation...
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Answer 1

Answer:

$\frac{dy}{dx} - \frac{y(x + 1)}{x} = 0 \\ \\ \therefore \: \frac{dy}{dx} = \frac{y(x + 1)}{x} \\ \\ \therefore \: \frac{1}{y} \frac{dy}{dx} = \frac{(x + 1)}{x} \\ \\ \therefore \: \frac{1}{y}dy = \frac{(x + 1)}{x} dx \\ \\ \therefore \: \frac{1}{y}dy = (1+ \frac{1}{x} ) dx \\ \\ integrating \: both \: sides \\ \\ \int\frac{1}{y}dy = \int(1+ \frac{1}{x} ) dx \\ \\\therefore \: log \: y = x + log \: x \: + c \\ \\ \therefore \: log \: y - log \: x = x + c \\ \\ \therefore \: log \: \frac{y}{x} = x + c \\ \\ \implies \: \frac{y}{x} = {e}^{x + c} \\ \\ \huge \red{\boxed{\implies \: y = x{e}^{x + c} }}$