$\text{Degree of the differential equation}\frac{d^{2}y}{d{x}^2}+3{\left(\frac{dy}{dx}\right)}^{\mathrm{1/42}}=y^2+e^x\text{is:}$

The correct option is 3.

To find the degree of the given differential equation, we must first make it free from radical signs and fractional powers of the derivatives. Let the given differential equation be:
$$\frac{d^{2}y}{d{x}^2}+3{\left(\frac{dy}{dx}\right)}^{1/3}=y^2+e^x$$

Step 1: Identify the order of the differential equation
The order of a differential equation is the order of the highest derivative present in the equation. In this equation, the highest derivative is the second derivative, $\frac{d^{2}y}{d{x}^2}$. Therefore, the order of the differential equation is 2.

Step 2: Eliminate the fractional exponent to find the degree
The degree of a differential equation is the exponent of the highest-order derivative when the equation is expressed as a polynomial in its derivatives. Currently, the first derivative term has a fractional power of 1/3.
To eliminate this fraction, we isolate the term with the fractional power on one side of the equation:
$$3{\left(\frac{dy}{dx}\right)}^{1/3}=y^2+e^x-\frac{d^{2}y}{d{x}^2}$$

Now, we raise both sides of the equation to the power of 3:
$$3^3\left(\frac{dy}{dx}\right)={(y^2+e^x-\frac{d^{2}y}{d{x}^2})}^3$$

Step 3: Determine the power of the highest order derivative
On expanding the right-hand side of the equation as a polynomial in the derivatives, the highest order derivative term $\frac{d^{2}y}{d{x}^2}$ will be raised to a maximum exponent of 3.
Since the highest power of the highest order derivative in this polynomial form is 3, the degree of the differential equation is 3.