Question

The degree of the differential equation \(7\left(\frac{d^3y}{dx^3}\right)^2 + 5\left(\frac{d^2y}{dx^2}\right)^3 + x\frac{dy}{dx} + y = 0\) will be

• A. 3
• B. 2
• C. 6
• D. 5

Hint:

Do not get confused between the order and the degree. The order is determined by the highest derivative (e.g., \(d^3y/dx^3\)), while the degree is the power of that highest derivative. A common mistake is to pick the highest power in the entire equation (which is 3 in this case, on the second derivative), but the degree is only related to the highest *order* derivative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the degree of a given differential equation. The degree is defined as the highest power (or exponent) of the highest-order derivative in the equation, after the equation has been cleared of any radicals and fractions in its derivatives.
Step 2: Key Formula or Approach:
1. Identify the highest-order derivative in the differential equation. This gives the order of the equation.
2. Identify the power (exponent) of this highest-order derivative. This power is the degree of the equation.
Step 3: Detailed Explanation:
The given differential equation is: \[ 7\left(\frac{d^3y}{dx^3}\right)^2 + 5\left(\frac{d^2y}{dx^2}\right)^3 + x\frac{dy}{dx} + y = 0 \] 1. Find the highest-order derivative:
The derivatives present in the equation are \(\frac{d^3y}{dx^3}\) (third order), \(\frac{d^2y}{dx^2}\) (second order), and \(\frac{dy}{dx}\) (first order).
The highest order among these is 3. So, the order of the differential equation is 3.
2. Find the degree:
The degree is the power of the highest-order derivative, which is \(\frac{d^3y}{dx^3}\).
The term containing the highest-order derivative is \(7\left(\frac{d^3y}{dx^3}\right)^2\).
The power of \(\frac{d^3y}{dx^3}\) in this term is 2.
Therefore, the degree of the differential equation is 2.
Step 4: Final Answer:
The degree of the given differential equation is 2.