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(A) The given differential equation is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = -\frac{1}{x}$ and $Q = x^2$.Firs

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$y = \frac{x^3}{3} + Cx$

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(A) The given differential equation is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = -\frac{1}{x}$ and $Q = x^2$.First,we find the Integrating Factor $(I.F.)$:$I.F. = e^{\int P dx} = e^{\int -\frac{1}{x} dx} = e^{-\log x} = e^{\log(x^{-1})} = \frac{1}{x}$.The general solution is given by $y(I.F.) = \int (Q 	imes I.F.) dx + C$.Substituting the values:$y \cdot \frac{1}{x} = \int (x^2 \cdot \frac{1}{x}) dx + C$$\frac{y}{x} = \int x dx + C$$\frac{y}{x} = \frac{x^2}{2} + C$$y = \frac{x^3}{2} + Cx$.

Find the general solution of the differential equation: $\frac{dy}{dx} - \frac{y}{x} = x^2$

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