Understanding the Degree of a Differential Equation

To find the degree of a differential equation, we first need to understand what order and degree mean in this context. The order of a differential equation is the order of the highest derivative present in the equation. The degree of a differential equation, provided it can be expressed as a polynomial in the derivatives, is the power of the highest order derivative.

Steps to Determine the Degree of a Differential Equation

Follow these steps to correctly identify the degree:

1. Examine the given differential equation and identify all the derivatives present.
2. Determine the highest order of these derivatives. This gives you the order of the differential equation.
3. Check if the differential equation is a polynomial equation in terms of its derivatives. This means that the derivatives (\(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\), etc.) should not be inside trigonometric functions, exponential functions, logarithms, or raised to fractional or negative powers. Functions of the independent variable (like \(\sin x\) or \(e^x\)) multiplying derivatives are acceptable.
4. If the equation is a polynomial in its derivatives, look at the term containing the highest order derivative.
5. The power (exponent) of this highest order derivative is the degree of the differential equation.
6. If the equation cannot be expressed as a polynomial in its derivatives (e.g., includes terms like \(\sin(\frac{dy}{dx})\) or \(e^{\frac{d^2y}{dx^2}})\), the degree is undefined.

Analyzing the Given Differential Equation

Let's apply these steps to the given differential equation:

\[{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} + {\left( {\frac{{dy}}{{dx}}} \right)^2} + \sin x\left( {\frac{{dy}}{{dx}}} \right) + y = 0\]

Step 1: Identify derivatives.

The derivatives present are \(\frac{{dy}}{{dx}}\) and \(\frac{{{d^2}y}}{{d{x^2}}}\).

Step 2: Determine the highest order.

The highest order derivative is \(\frac{{{d^2}y}}{{d{x^2}}}\), which is a second-order derivative. So, the order of this differential equation is 2.

Step 3: Check for polynomial form in derivatives.

Let's look at the terms involving derivatives:

• \({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3}\): This is the second derivative raised to the power of 3. This is a polynomial term in the derivative.
• \({\left( {\frac{{dy}}{{dx}}} \right)^2}\): This is the first derivative raised to the power of 2. This is a polynomial term in the derivative.
• \(\sin x\left( {\frac{{dy}}{{dx}}} \right)\): This is the first derivative multiplied by \(\sin x\). \(\sin x\) is a function of the independent variable \(x\), not the derivative itself. This is a polynomial term in the derivative.

The equation contains only derivatives raised to positive integer powers, multiplied by constants or functions of \(x\). Therefore, the differential equation is a polynomial in its derivatives.

Step 4 & 5: Find the power of the highest order derivative.

The highest order derivative is \(\frac{{{d^2}y}}{{d{x^2}}}\). In the term \({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3}\), this highest order derivative is raised to the power of 3. This is the highest power among all terms involving the highest order derivative.

Therefore, the degree of the differential equation is 3.

Conclusion: The Degree of the Differential Equation

Based on our analysis, the highest order derivative is \(\frac{{{d^2}y}}{{d{x^2}}}\) (order 2), and its power in the polynomial form of the equation is 3. Thus, the degree of the differential equation \({\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^3} + {\left( {\frac{{dy}}{{dx}}} \right)^2} + \sin x\left( {\frac{{dy}}{{dx}}} \right) + y = 0\) is 3.

This aligns with one of the given options.