The solution of the differential equation 2 x | vedclass

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(B) Given differential equation is $2 x \frac{d y}{d x} - y = 4$. Rearranging the terms,we get $2 x \frac{d y}{d x} = 4 + y$. Separating the variables

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Parabolas

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(B) Given differential equation is $2 x \frac{d y}{d x} - y = 4$. Rearranging the terms,we get $2 x \frac{d y}{d x} = 4 + y$. Separating the variables,we have $\frac{2}{4 + y} d y = \frac{1}{x} d x$. Integrating both sides,$\int \frac{2}{4 + y} d y = \int \frac{1}{x} d x$. This gives $2 \ln |4 + y| = \ln |x| + C$,where $C = \ln |c|$. Using logarithmic properties,$\ln (4 + y)^2 = \ln |c x|$. Taking the exponential of both sides,$(4 + y)^2 = c x$. This equation is of the form $(y - k)^2 = 4 a (x - h)$,which represents a family of parabolas.

The solution of the differential equation $2 x \left(\frac{d y}{d x}\right) - y = 4$ represents a family of

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