The solution of the differential equation 2x | vedclass

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(C) Given differential equation: $2x \frac{dy}{dx} - y = 3$Rearranging the terms: $2x \frac{dy}{dx} = y + 3$Separating the variables: $\frac{dy}{y + 3

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parabolas

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(C) Given differential equation: $2x \frac{dy}{dx} - y = 3$Rearranging the terms: $2x \frac{dy}{dx} = y + 3$Separating the variables: $\frac{dy}{y + 3} = \frac{dx}{2x}$Integrating both sides: $\int \frac{dy}{y + 3} = \int \frac{dx}{2x}$$\ln|y + 3| = \frac{1}{2} \ln|x| + C$Multiply by $2$: $2 \ln|y + 3| = \ln|x| + 2C$$\ln(y + 3)^2 = \ln|x| + C'$$(y + 3)^2 = kx$, where $k = e^{C'}$This equation is of the form $(y - k_1)^2 = 4a(x - k_2)$, which represents a family of parabolas.

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

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