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(A) The given differential equation is $x \frac{dy}{dx} + 2y = x^2$  $(1)$.Dividing both sides by $x$,we get $\frac{dy}{dx} + \frac{2}{x}y = x$.This i

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$y = \frac{x^2}{4} + Cx^{-2}$

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(A) The given differential equation is $x \frac{dy}{dx} + 2y = x^2$  $(1)$.Dividing both sides by $x$,we get $\frac{dy}{dx} + \frac{2}{x}y = x$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = \frac{2}{x}$ and $Q = x$.The integrating factor $(I.F.)$ is given by $I.F. = e^{\int P dx} = e^{\int \frac{2}{x} dx} = e^{2 \ln|x|} = e^{\ln(x^2)} = x^2$.The general solution is $y \cdot (I.F.) = \int Q \cdot (I.F.) dx + C$.Substituting the values,we get $y \cdot x^2 = \int x \cdot x^2 dx + C$.$y \cdot x^2 = \int x^3 dx + C$.$y \cdot x^2 = \frac{x^4}{4} + C$.Dividing by $x^2$,we obtain $y = \frac{x^2}{4} + Cx^{-2}$.

Find the general solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ where $x \neq 0$.

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