1. Given differential equation

   $\left(1+3\frac{dy}{dx}\right)^{2/3}=4\frac{d^3y}{dx^3}$

2. Find the order

   The order of a differential equation is the order of the highest derivative present.

   Here, the derivatives present are:

   • $\dfrac{dy}{dx}$
   • $\dfrac{d^3y}{dx^3}$

   The highest order derivative is $\dfrac{d^3y}{dx^3}$.

   Hence, the order is

   $3$

3. Find the degree

   The degree is defined only when the differential equation is a polynomial in derivatives after removing radicals/fractional powers of derivatives.

   First, remove the fractional power by cubing both sides:

   $\left(1+3\frac{dy}{dx}\right)^2=\left(4\frac{d^3y}{dx^3}\right)^3$

   $\left(1+3y'\right)^2=64\left(y'''\right)^3$

   Now the equation is polynomial in derivatives.

   The highest order derivative is $y'''$, and its highest power is $3$.

   Hence, the degree is

   $3$

4. Final answer

   $\boxed{(\text{order},\text{degree})=(3,3)}$

5. Option check

   • A: $(1, \tfrac{2}{3})$ ❌
   • B: $(3,1)$ ❌
   • C: $(3,3)$ ✅
   • D: $(1,2)$ ❌

So, the correct option is C.