Step 1: Understanding the Concept:
The degree of a differential equation is the power of the highest-order derivative term, provided the equation is a polynomial in its derivatives. The first step is to clear any fractions or radicals involving the dependent variable or its derivatives.
Step 2: Key Formula or Approach:
1. Identify the highest-order derivative in the equation. This determines the order of the equation.
2. Rewrite the equation to eliminate any denominators or fractional powers involving derivatives.
3. The degree is the highest integer power of the highest-order derivative term in the resulting polynomial equation.
Step 3: Detailed Explanation:
The given differential equation is:
\[ y''' + y = \frac{5}{y^3} \]
The equation is not in polynomial form due to the term $\frac{5}{y^3}$. However, the definition of degree is concerned with the derivatives being in polynomial form, not necessarily the dependent variable y itself.
The derivatives present are $y'''$ and $y$. The highest-order derivative is $y'''$ (the third derivative).
So, the order of the differential equation is 3.
Now, we look at the power of this highest-order derivative, $y'''$.
In the term $y'''$, the power is 1.
The equation is already a polynomial in terms of its derivatives (like $y', y'', y'''$). There is no need to manipulate the equation further to determine the degree.
The highest order derivative is $y'''$, and its power is 1.
Step 4: Final Answer:
The degree of the differential equation is 1. Therefore, option (A) is correct.