The solution of the differential equation 2x | vedclass

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(B) The given differential equation is $2x \frac{dy}{dx} - y = 0$. Rearranging the terms,we get $2x \frac{dy}{dx} = y$. Separating the variables,we ha

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Parabola

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(B) The given differential equation is $2x \frac{dy}{dx} - y = 0$. Rearranging the terms,we get $2x \frac{dy}{dx} = y$. Separating the variables,we have $\frac{dy}{y} = \frac{dx}{2x}$. Integrating both sides,$\int \frac{dy}{y} = \frac{1}{2} \int \frac{dx}{x}$,which gives $\ln|y| = \frac{1}{2} \ln|x| + C$. This can be written as $\ln|y| = \ln|x^{1/2}| + C$,so $y = k \sqrt{x}$,where $k = e^C$. Using the condition $y(1) = 2$,we get $2 = k \sqrt{1}$,so $k = 2$. The solution is $y = 2 \sqrt{x}$,which implies $y^2 = 4x$. This equation represents a parabola.

The solution of the differential equation $2x \frac{dy}{dx} - y = 0$,with the initial condition $y(1) = 2$,represents which of the following curves?

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