Let y = y(x) be the solution curve of the differential equation, \left(\right. y^{2} - x \left.\right) \frac{dy}{dx} = 1, satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is:

Engineering

Mathematics

Linear Differential Equation

Question

Let y = y(x) be the solution curve of the differential equation, $(y^{2} - x) \frac{dy}{dx} = 1$ , satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is:

2 + e

2 – e

–e

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$\frac{dy}{dx} + x = y^{2}$

$I . F . = e^{\int l . dy} = e^{y}$

$x . e^{y} = \int y^{2} . e^{y} . dy = y^{2} . e^{y} - \int 2 y . e^{y} . dy$

⇒ y²e^y – 2(y.e^y –e^y) + c

x.e^y = y²e^y –2ye^y + 2e^y + C

x = y² –2y + 2 + c. e^–y

x = 0, y = 1

$0 = 1 - 2 + 2 + \frac{c}{e}$

c = –e

y = 0, x = 0 – 0 + 2 + (–e) (e^–0)

x = 2 –e