The solution of the differential equation 2x | vedclass

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(C) Step $1$: Rewrite the differential equation as a linear differential equation: $\frac{dy}{dx} - \frac{1}{2x}y = \frac{3}{2x}$.Step $2$: Find the i

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parabolas

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(C) Step $1$: Rewrite the differential equation as a linear differential equation: $\frac{dy}{dx} - \frac{1}{2x}y = \frac{3}{2x}$.Step $2$: Find the integrating factor $IF = e^{\int -\frac{1}{2x} dx} = e^{-\frac{1}{2} \ln|x|} = x^{-1/2} = \frac{1}{\sqrt{x}}$.Step $3$: Multiply the equation by $IF$: $\frac{d}{dx} (y \cdot x^{-1/2}) = \frac{3}{2x} \cdot x^{-1/2} = \frac{3}{2} x^{-3/2}$.Step $4$: Integrate both sides: $y \cdot x^{-1/2} = \int \frac{3}{2} x^{-3/2} dx = \frac{3}{2} \cdot \frac{x^{-1/2}}{-1/2} + C = -3x^{-1/2} + C$.Step $5$: Multiply by $\sqrt{x}$ to get $y = -3 + C\sqrt{x}$, which can be rearranged as $y + 3 = C\sqrt{x}$. Squaring both sides gives $(y+3)^2 = C^2 x$, 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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