First slide

Types of solutions of differential equations

Question

$The x-intercept of the tangent to a curve is equal to the coordinate of the point of contact.$
$The equation of the curve through the point (1,1) is$

Moderate

A

${ye}^{x/y} =e$
B

${xe}^{x/y} =e$
C

${xe}^{y/x} =e$
D

${ye}^{y/x} =e$

Solution

$\begin{array}{l} equation of tangent Y - y = \frac{dy}{dx} (x - x) \\ For X-intercept Y = 0 \to x = x - y \frac{d x}{d y} \\ According to the question x - y \frac{d x}{d y} = y \to \frac{d y}{d x} = \frac{y}{x - y} put y = V x \\ \to x \frac{dv}{dx} = \frac{v^{2}}{1 - v} \to \int \frac{1 - v}{v^{2}} dv = \int \frac{dx}{x} \\ \Rightarrow \frac{- 1}{V} - \log V = \log x + c \end{array}$
$\begin{array}{l} \Rightarrow - x / y - \log y + \log x = \log x + c \\ \Rightarrow - \frac{x}{y} = \log y + c pass (1, 1) \Rightarrow c = - 1 \\ \Rightarrow y = e \cdot e^{- x / y} \Rightarrow e^{x / y} = \frac{e}{y} \Rightarrow y e^{x / y} = e \end{array}$

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