The differential equation of all parabolas having axis parallel to y-axis: (A) \(\left(\mathrm{d}^{3} \mathrm{x} / \mathrm{dy}^{3}\right)^{2}=0\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)=0\) (C) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\)

To find the derivatives, let's first label our equation for easier differentiation: \(x = a(y-k)^2 + h\) Now, let's find the first and second derivatives with respect to y: 1st derivative: \(\frac{dx}{dy} = 2a(y-k)\) 2nd derivative: \(\frac{d^2x}{dy^2} = 2a\) Notice that the 2nd derivative is a constant value (2a).

We will now go through each of the given options to see which one fits the derived expressions: (A) \(\left(\frac{d^3 x}{dy^3}\right)^2 = 0\) The third derivative would be 0, as the second derivative is constant. This option may be a correct answer. (B) \(\left(\frac{d^3 y}{dx^3}\right)=0\) This choice has a third derivative of y with respect to x, which isn't applicable here since we are differentiating x with respect to y. So this option is incorrect. (C) \(\left(\frac{d^3 y}{dx^3}\right)+\left(\frac{d^2 y}{dx^2}\right)=0\) Similarly to option B, this option is also incorrect since it deals with the derivation of y with respect to x. (D) \(\left(\frac{d^2 y}{dx^2}\right)=0\) This choice has the second derivative of y with respect to x, which isn't applicable here since we are differentiating x with respect to y. So this option is also incorrect. Since we have eliminated options B, C, and D, the correct answer is: \( \boxed{\text{(A)}\ \left(\frac{d^3 x}{dy^3}\right)^2 = 0} \)