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Q.

If $\frac{d y}{d x} = \frac{x y + y}{x y + x}$ , then the solution of the differential equation is

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a

$y = x e^{x} + c$

b

$y = e^{x} + c$

c

$y = A x e^{x - y}$

d

$y = x + A$

answer is C.

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Detailed Solution

$\begin{array}{l} ∵ \frac{d y}{d x} = \frac{y (x + 1)}{x (y + 1)} \\ \Rightarrow \int \frac{y + 1}{y} d y = \int \frac{x + 1}{x} d x \\ \Rightarrow \int d y + \int \frac{1}{y} d y = \int d x + \int \frac{1}{x} d x \\ \Rightarrow y + \ln y = x + \ln x + c \\ \Rightarrow \ln (\frac{y}{x}) = x - y + c \Rightarrow \frac{y}{x} = e^{x - y + c} \\ \Rightarrow y = (x e^{x - y}) e^{c} \Rightarrow y = A x e^{x - y} \end{array}$

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