The differential equation x frac{dy}{dx} = 2y | vedclass

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(B) Given the differential equation: $x \frac{dy}{dx} = 2y$. Separating the variables,we get: $\frac{dy}{y} = 2 \frac{dx}{x}$. Integrating both sides:

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a family of parabolas with vertex at the origin and axis along the positive $Y$-axis.

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(B) Given the differential equation: $x \frac{dy}{dx} = 2y$. Separating the variables,we get: $\frac{dy}{y} = 2 \frac{dx}{x}$. Integrating both sides: $\int \frac{dy}{y} = 2 \int \frac{dx}{x}$. This gives: $\ln|y| = 2 \ln|x| + C$. Using properties of logarithms: $\ln|y| = \ln|x^2| + C$. Taking the exponential of both sides: $y = e^{\ln|x^2| + C} = e^C \cdot x^2$. Let $e^C = k$,where $k$ is an arbitrary constant. Thus,$y = kx^2$. This equation represents a family of parabolas with the vertex at the origin $(0,0)$ and the axis along the $Y$-axis.

The differential equation $x \frac{dy}{dx} = 2y$ represents ....

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