To determine the degree of the given differential equation, let's first analyze the equation:
\(x = \frac{dy}{dx} + \frac{1}{2!} \left(\frac{dy}{dx}\right)^2 + \frac{1}{3!} \left(\frac{dy}{dx}\right)^3 + \cdots\)
This equation is an infinite series expansion, where each term is a multiple or power of the derivative \(\frac{dy}{dx}\).
Step 1: Identify the Highest Derivative Order
The given series appears to be similar to the expansion of the exponential function, specifically the expansion of \(e^x\), but instead, we have \(\frac{dy}{dx}\) as the variable:
\(e^{\frac{dy}{dx}} = 1 + \frac{\frac{dy}{dx}}{1!} + \frac{\left(\frac{dy}{dx}\right)^2}{2!} + \frac{\left(\frac{dy}{dx}\right)^3}{3!} + \cdots\)
Comparing this with the given equation, it’s implicit that the equation can be restructured:
\(x \approx e^{\frac{dy}{dx}} - 1\)
or equivalently:
\(e^{\frac{dy}{dx}} = x + 1\)
Step 2: Simplifying and Solving for the Order
Rewrite the expression without the approximation for the greatest power term, we assume
From \(e^z = x + 1\), solving the highest degree seen if possible from the simplified view:
Fire the original form again for literal definition purposes and squaring restrictions for a single derivative representative to seek for highest-degree terms:
\(0 = \left(\frac{dy}{dx}\right)^3 = 0\), where the theoretically described highest power is 1 upon the implicit rewriting or rationale visit.
Conclusion: Since the given equation is already expressed by its nature as \(\frac{dy}{dx}\) in the first order power context, the highest degree of a derivative is inherently 1.
Thus, the degree of this differential equation is 1, as there is no greater power of the derivative involved when observed in its operational form.