Question

If \( \dfrac{d^2y}{dx^2} = \cos \left(\dfrac{dy}{dx}\right) \), find the order and the degree of the resulting differential equation.

• A. Order \(2\), Degree \(4\)
• B. Cannot be determined
• C. Order \(3\), Degree \(1\)
• D. Data Insufficient
• E. None of these

Hint:

Degree of a differential equation exists only when the equation can be written as a polynomial in derivatives. If derivatives appear inside functions like \(\sin\), \(\cos\), or \(\log\), the degree is not defined.

Answer 1

Answer: (E)

Concept:
• Order of a differential equation is the highest order derivative present in the equation.
• Degree is defined only when the differential equation can be expressed as a polynomial in derivatives.
• If derivatives appear inside transcendental functions such as \(\sin, \cos, e^x, \log\), the degree is not defined.

Step 1: Identify the highest order derivative. The given equation is \[ \frac{d^2y}{dx^2} = \cos\left(\frac{dy}{dx}\right) \] The highest derivative present is \[ \frac{d^2y}{dx^2} \] Thus, the order is \[ 2 \]

Step 2: Determine the degree. The first derivative \( \frac{dy}{dx} \) appears inside the cosine function, which is a transcendental function. Hence, the differential equation cannot be expressed as a polynomial in derivatives. Therefore, the degree is not defined.

Step 3: Select the correct option. Order \(=2\) but degree not defined. Since none of the options match this exactly, the correct choice is \[ \text{None of these} \]