The curve represented by the differential equation $x d y - y d x = y d y$ , intersects the y-axis at $A (0, 1)$ and the line $y = e$ at $(a, b,)$ , then the value of $(a + b)$ is ____________

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Solution

$\begin{matrix} x d y - y d x = y d y \frac{x d y - y d x}{y^{2}} = \frac{d y}{y} \frac{y d x - x d y}{y^{2}} = - \frac{d y}{y} d (\frac{x}{y}) = \frac{- d y}{y} \frac{x}{y} = - log y + C \frac{x}{y} = log (\frac{1}{y}) + C 0 = 0 + C C = 0 \frac{x}{y} = log (\frac{1}{y}) p u t t i n g y = e \frac{x}{e} = - 1 x = - e (- e, e) a + b = 0 \end{matrix}$

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The curve represented by the differential equation xdy−ydx=ydy, intersects the y-axis at A(0,1) and the line y=e at (a,b,), then the value of (a+b) is ____________

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The curve represented by the differential equation $x d y - y d x = y d y$ , intersects the y-axis at $A (0, 1)$ and the line $y = e$ at $(a, b,)$ , then the value of $(a + b)$ is ____________