Question

Determine the order and degree of the differential equation \( \frac{d^3y}{dx^3} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} \):

• A. \(5, 2\)
• B. \(3, 5\)
• C. \(3, 2\)
• D. \(2, 3\)

Hint:

When determining the order and degree of a differential equation, focus on the highest derivative and ensure the equation is in a form free from radicals and fractional powers to define the degree.

Answer 1

The Correct Option is B

To determine the order and degree of the given differential equation, follow these steps: 
Step 1: {Order of the differential equation} The order of a differential equation is the highest derivative that appears in the equation. In this case, the highest derivative is \( \frac{d^3y}{dx^3} \), which indicates that the order of the differential equation is 3. 
Step 2: {Degree of the differential equation} The degree of the differential equation is the power of the highest derivative after clearing any radicals and fractional powers. In the given equation, the highest derivative is \( \frac{d^3y}{dx^3} \), which appears with a fractional power of \( \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} \). To clear the fractional exponent, raise both sides of the equation to the power of \( \frac{2}{5} \), which will remove the fractional power and give a polynomial form. The degree of the differential equation is the exponent of the highest derivative after this operation. The degree becomes 5, as the exponent \( \frac{5}{2} \) is cleared to become 5. Thus, the order is 3, and the degree is 5.