The general solution of the differential equa | vedclass

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(D) Given the differential equation: $\frac{dy}{dx} = \frac{y + \sqrt{x^2 - y^2}}{x} \dots (i)$ Since this is a homogeneous differential equation,we s

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$\sin^{-1}\left(\frac{y}{x}\right) = \log x + c$,where $c$ is a constant of integration.

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(D) Given the differential equation: $\frac{dy}{dx} = \frac{y + \sqrt{x^2 - y^2}}{x} \dots (i)$ Since this is a homogeneous differential equation,we substitute $y = vx$,which implies $\frac{dy}{dx} = v + x\frac{dv}{dx} \dots (ii)$ Substituting $(ii)$ into $(i)$: $v + x\frac{dv}{dx} = \frac{vx + \sqrt{x^2 - v^2x^2}}{x}$ $v + x\frac{dv}{dx} = v + \sqrt{1 - v^2}$ $x\frac{dv}{dx} = \sqrt{1 - v^2}$ Separating the variables: $\int \frac{dv}{\sqrt{1 - v^2}} = \int \frac{dx}{x}$ Integrating both sides: $\sin^{-1}(v) = \log|x| + c$ Substituting $v = \frac{y}{x}$ back: $\sin^{-1}\left(\frac{y}{x}\right) = \log|x| + c$

The general solution of the differential equation $\frac{dy}{dx} = \frac{y + \sqrt{x^2 - y^2}}{x}$ is

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