The solution of \(\mathrm{y}^{5} \mathrm{x}+\mathrm{y}-\mathrm{x}(\mathrm{dy} / \mathrm{dx})=0\) is: (A) \((\mathrm{x} / \mathrm{y})^{5}+\left(\mathrm{x}^{4} / 4\right)=\mathrm{c}\) (B) \((\mathrm{xy})^{4}+\left(\mathrm{x}^{5} / 5\right)=\mathrm{c}\) (C) \(\left(x^{5} / 5\right)+(1 / 4)(x / y)^{4}=c\) (D) \(\left(x^{4} / y\right)+(1 / 5)(x / y)^{5}=c\)

The given differential equation is: \[y^5 x + y - x \frac{dy}{dx} = 0\]

We are given the possible solution as: \[\frac{x}{y}^5 + \frac{x^4}{4} = c\] Now we will differentiate both sides with respect to x. \[\frac{d}{dx}\left(\frac{x}{y}^5 + \frac{x^4}{4}\right) = \frac{d}{dx}(c)\] Since c is a constant, its derivative with respect to x is 0. \[\frac{d}{dx}\left(\frac{x}{y}^5 + \frac{x^4}{4}\right) = 0\] Calculate the derivative: \[\frac{d\left(\frac{x}{y}\right)^5}{dx} + \frac{d\left(\frac{x^4}{4}\right)}{dx}=0\] Using chain rule and product rule for derivatives, \[5\left(\frac{x}{y}\right)^4 \left(\frac{dy}{dx}\frac{x}{y^2} - \frac{y}{x^2}\right) + x^3 = 0\] Now see if this equation is equivalent to the given differential equation. They are not equivalent, so option (A) is incorrect.

We are given the possible solution as: \[(xy)^4 + \frac{x^5}{5} = c\] This equation seems unusual because it does not contain the term \(\frac{dy}{dx}\), so it doesn't make sense to further proceed with this option. Hence, option (B) is incorrect.

We are given the possible solution as: \[\frac{x^5}{5} + \frac{1}{4}(x / y)^4 = c\] Now we will differentiate both sides with respect to x. \[\frac{d}{dx}\left(\frac{x^5}{5} + \frac{1}{4}(x / y)^4\right) = \frac{d}{dx}(c)\] Since c is a constant, its derivative with respect to x is 0. \[\frac{d}{dx}\left(\frac{x^5}{5} + \frac{1}{4}(x / y)^4\right) = 0\] Calculate the derivative: \[x^4 - x^3\frac{dy}{dx}\frac{1}{y^5} = 0\] Now see if this equation is equivalent to the given differential equation. They are not equivalent, so option (C) is incorrect.

We are given the possible solution as: \[\frac{x^4}{y} + \frac{1}{5}(x / y)^5 = c\] Now we will differentiate both sides with respect to x. \[\frac{d}{dx}\left(\frac{x^4}{y} + \frac{1}{5}(x / y)^5\right) = \frac{d}{dx}(c)\] Since c is a constant, its derivative with respect to x is 0. \[\frac{d}{dx}\left(\frac{x^4}{y} + \frac{1}{5}(x / y)^5\right) = 0\] Calculate the derivative: \[x^3\left(\frac{dy}{dx}\frac{1}{y^2} - \frac{1}{x^4}\right) + \frac{dy}{dx}\frac{x^4}{5y^5} = 0\] Now see if this equation is equivalent to the given differential equation. \[y^5x + y - x\frac{dy}{dx} = 0\] They are equivalent, so option (D) is the correct solution.