Step 1: Understanding the Concept:
The structural classification of a differential equation is defined by two fundamental properties:
1. Order: The order is the highest derivative present anywhere within the differential equation.
2. Degree: The degree is the power exponent of the highest-order derivative component, provided the differential equation is expressed as a polynomial with respect to all its differential derivatives (meaning all derivatives must be free from fractional exponents, radical signs, or transcendental arguments).
Step 2: Detailed Explanation:
Let's analyze the terms within the given differential equation:
$$ \left(\frac{d^3y}{dx^3}\right)^2 + 4\left(\frac{dy}{dx}\right)^4 + y = \sin(x) $$
1. Determine the Order: Look at the derivative terms appearing in the equation:
- There is a first derivative term: $\frac{dy}{dx}$
- There is a third derivative term: $\frac{d^3y}{dx^3}$
The highest number of times the function has been differentiated is 3. Therefore, the Order = 3.
2. Determine the Degree:
First, verify if the equation is a proper polynomial in terms of its derivatives. Since all individual derivative terms ($\frac{d^3y}{dx^3}$ and $\frac{dy}{dx}$) are raised to positive integer exponents ($2$ and $4$) and are not trapped inside functions like $\sin(\frac{dy}{dx})$ or $e^{\frac{dy}{dx}}$, the degree is valid.
Now, isolate the highest-order derivative term, which is $\left(\frac{d^3y}{dx^3}\right)^2$. Look at the exponent power attached to this specific term. It is raised to the power of 2. Therefore, the Degree = 2.
The differential equation has an Order of 3 and a Degree of 2, which matches option (B).
Step 3: Final Answer:
The order and degree of the differential equation are Order 3, Degree 2.