The general solution of the differential equa | vedclass

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(B) The given differential equation is $2x \frac{dy}{dx} - y = 3$. Rearranging the terms,we get $2x \frac{dy}{dx} = y + 3$. Separating the variables,w

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parabolas

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(B) The given differential equation is $2x \frac{dy}{dx} - y = 3$. Rearranging the terms,we get $2x \frac{dy}{dx} = y + 3$. Separating the variables,we have $\frac{dy}{y+3} = \frac{dx}{2x}$. Integrating both sides,we get $\int \frac{dy}{y+3} = \frac{1}{2} \int \frac{dx}{x}$. This gives $\ln|y+3| = \frac{1}{2} \ln|x| + C_1$. Multiplying by $2$,we get $2 \ln|y+3| = \ln|x| + 2C_1$. Using properties of logarithms,$\ln(y+3)^2 = \ln|x| + \ln|c|$,where $c = e^{2C_1}$. Thus,$(y+3)^2 = cx$,which is the equation of a family of parabolas.

The general solution of the differential equation $2x \frac{dy}{dx} - y = 3$ is a family of

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