Question

The order and degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = a^x \) are ............. respectively.

• A. 1, 1
• B. 1, 2
• C. 2, 2
• D. 2, 1

Hint:

- {Order:} Look for the highest derivative (e.g., \( y' \), \( y'' \), \( y''' \)). - {Degree:} Look for the exponent of the highest derivative. Ensure the equation is a polynomial in its derivatives before determining the degree.

Answer 1

The Correct Option is C

Step 1: Determine the order of the differential equation. The order is the order of the highest derivative present. The highest derivative is \( \frac{d^2y}{dx^2} \), which is the second derivative. Therefore, the order is 2.
Step 2: Determine the degree of the differential equation. The degree is the highest power of the highest-order derivative, after the equation has been cleared of radicals and fractions with respect to its derivatives. The equation is already a polynomial in its derivatives. The highest-order derivative is \( \frac{d^2y}{dx^2} \), and its power is 2. Therefore, the degree is 2.