Which of the following differential equations has y = x as one of its particular solution?

Option (C) is correct.

The given equation of curve is y=x Differentiating with respect to x, we get :

                       $\frac{dy}{dx}=1$     ….(i)

Again, differentiating with respect to x, we get :

$\frac{d^{2}y}{d{x}^2}=0$………..(ii)

Now, on substituting the values of $y,\frac{d^{2}y}{d{x}^2}$ and $\frac{dy}{dx}$ from equation (i) and (ii) in each of the given alternatives, we find that only the differential equation given in alternative $C$ is correct.

$\begin{array}{r}\frac{d^{2}y}{d{x}^2}-x^2\frac{dy}{dx}+xy=0-x^2\cdot 1+x\cdot x\\ =-x^2+x^2\\ =0\end{array}$