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(D) Given the differential equation: $x \frac{dy}{dx} + 2y = x^2$ Divide by $x$ to get the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$: $\frac

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

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(D) Given the differential equation: $x \frac{dy}{dx} + 2y = x^2$ Divide by $x$ to get the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$: $\frac{dy}{dx} + \frac{2}{x}y = x$ Here,$P(x) = \frac{2}{x}$ and $Q(x) = x$. The Integrating Factor ($I$.$F$.) is given by: $I.F. = e^{\int P(x) dx} = e^{\int \frac{2}{x} dx} = e^{2 \ln|x|} = e^{\ln(x^2)} = x^2$ Multiplying the differential equation by the $I$.$F$. $(x^2)$: $x^2 \frac{dy}{dx} + 2xy = x^3$ This can be written as: $\frac{d}{dx}(y \cdot x^2) = x^3$ Integrating both sides with respect to $x$: $y \cdot x^2 = \int x^3 dx$ $y \cdot x^2 = \frac{x^4}{4} + C$ Dividing by $x^2$: $y = \frac{x^4 + 4C}{4x^2}$ Since $4C$ is an arbitrary constant,we can write it as $C$: $y = \frac{x^4 + C}{4x^2}$

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ is:

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