Question

Find the order of the differential equation \(\left(\frac{d^2 y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y^4 = 0\)

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Always look for the highest derivative symbol to find the order:
- \(\frac{dy}{dx}\) represents first order.
- \(\frac{d^2y}{dx^2}\) represents second order.
The power to which this highest derivative is raised gives the degree of the equation (once radical signs are removed).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To analyze a differential equation, we define two fundamental properties: its order and its degree.
The order of a differential equation is the order of the highest derivative appearing in the equation.
The degree is the power of the highest-order derivative when the equation is expressed as a polynomial in derivatives.

Step 2: Detailed Explanation:
Let us inspect the given differential equation:
\[ \left(\frac{d^2 y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y^4 = 0 \]
This equation contains three distinct terms involving the dependent variable \(y\) and its derivatives:
1. The first term is the second-order derivative, \(\frac{d^2 y}{dx^2}\), raised to the power of 2.
2. The second term is the first-order derivative, \(\frac{dy}{dx}\), raised to the power of 3.
3. The third term is the dependent variable \(y\) raised to the power of 4.
The highest-order derivative present in this entire equation is the second-order derivative, \(\frac{d^2 y}{dx^2}\).
Therefore, by definition, the order of this differential equation is 2.
Additionally, the exponent of this highest-order derivative is 2, which means the degree of this differential equation is also 2.
This matches Option (B).

Step 3: Final Answer:
The correct option is (B).