Question

The order of the differential equation \( \left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^5 = 0 \) is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 1 \)
• D. \( 5 \)
• E. \( 6 \)

Hint:

Always look for the "d" count in the notation. \( \frac{d^ny}{dx^n} \) tells you it's the \( n \)-th order. Don't be distracted by exponents like the 5 in \( (dy/dx)^5 \); they don't affect the order of the equation.

Answer 1

The Correct Option is A

Concept: In the study of differential equations, two fundamental properties are order and degree. The order of a differential equation is the order of the highest derivative present in the equation. For example, \( \frac{d^2y}{dx^2} \) is a second-order derivative. It is crucial to distinguish order from degree; the degree is the power of the highest-order derivative when the equation is a polynomial in its derivatives.

Step 1: Identify all derivatives in the equation.
We analyze the given differential equation: \[ \left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^5 = 0 \] The derivatives present are:
• \( \frac{d^3y}{dx^3} \), which is a third-order derivative.
• \( \frac{d^2y}{dx^2} \), which is a second-order derivative.
• \( \frac{dy}{dx} \), which is a first-order derivative.

Step 2: Determine the highest order.
The order of the differential equation is the highest derivative present. In this equation, the highest derivative is \( \frac{d^3y}{dx^3} \), which is the third derivative. Therefore, the order of the differential equation is 3. Note that the powers (2, 2, and 5) determine the degree and complexity, but not the order.