Question

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

• A. \( \text{Not defined} \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 1 \)

Hint:

Be careful not to look at the larger power of 3 on the left side of the equation. The degree is determined strictly by the exponent on the highest-order derivative (\( \frac{d^2y}{dx^2} \)), not by larger exponents attached to lower-order derivatives.

Answer 1

The Correct Option is B

Concept: The degree of a differential equation is the power index of its highest-order derivative component, calculated only after the equation has been cleared of any fractional exponents or radicals affecting its derivatives.

Step 1: Clear the fractional exponent from the equation structure.
The left side of the equation features a fractional power exponent of \( \frac{3}{2} \). To clear the denominator radical, square both sides of the equation: \[ \left( \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}} \right)^2 = \left( \frac{d^2y}{dx^2} \right)^2 \] This simplifies the expression to a clean polynomial format: \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = \left( \frac{d^2y}{dx^2} \right)^2 \]

Step 2: Identify the highest-order derivative and its exponent index.
Look across the transformed equation to find its derivative properties:

• The highest-order derivative present is the second derivative \( \frac{d^2y}{dx^2} \), which means the Order = 2.
• The power exponent attached directly to this highest-order derivative term is 2, which gives a Degree = 2.