The differential equation whose solution represents the family $x^2 y=4 e^x+c$, where $c$ is an arbitrary constant, is

Given the solution $x^2 y=4 e^x+c$, we eliminate the arbitrary constant $c$ through differentiation.

Differentiating both sides with respect to $x$ using the product rule for the left-hand side:

$\frac{\mathrm{d}}{\mathrm{d} x}(x^2 y) = x^2 \frac{\mathrm{d}y}{\mathrm{d} x} + 2x y$

The derivative of the right-hand side yields:

$\frac{\mathrm{d}}{\mathrm{d} x}(4e^x + c) = 4e^x$

Equating both derivatives:

$x^2 \frac{\mathrm{d}y}{\mathrm{d} x} + 2x y = 4e^x$

Rewriting in standard differential equation form:

$x^2 \frac{\mathrm{d}y}{\mathrm{d} x} + (2x y - 4e^x) = 0$

This corresponds to option B.

The differential equation is $\boxed{\text{B}}$