The order of a differential equation is the order of the highest derivative present in the equation. In the given equation $1+(\frac{dy}{dx})^{3}=\lambda(\frac{d^{3}y}{dx^{3}})^{2}$, the highest derivative is $\frac{d^{3}y}{dx^{3}}$. Therefore, the order is 3.
The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in derivatives. Here, the highest order derivative is $\frac{d^{3}y}{dx^{3}}$ and its exponent is 2. Thus, the degree is 2.
The question asks for the product of the order and the degree. $$Product = Order \times Degree$$ $$Product = 3 \times 2 = 6$$
Final Answer: 6
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