Step 1: Understanding the Concept:
The degree of a differential equation is the power of the highest order derivative when the equation is written as a polynomial in derivatives. Crucially, the equation must be free of radicals and fractions in the power of its derivatives.
In this equation, we see a fractional power of \(3/2\). This fractional power prevents us from reading the degree directly. We must first "rationalize" the equation by raising it to a power that turns all fractional exponents into integers.
Key Formula or Approach:
1. Identify the highest order derivative. Here, \(d^2y/dx^2\) (second-order).
2. Eliminate the denominator of the fractional exponent by squaring both sides.
Step 2: Detailed Explanation:
Given: \(\left[ 1 + \left(\frac{dy}{dx}\right)^{2} \right]^{3/2} = \frac{d^{2}y}{dx^{2}}\)
The highest order derivative present is \(\frac{d^{2}y}{dx^{2}}\). Thus, Order = 2.
To find the degree, square both sides to remove the \(1/2\) in the exponent:
\[ \left( \left[ 1 + \left(\frac{dy}{dx}\right)^{2} \right]^{3/2} \right)^{2} = \left( \frac{d^{2}y}{dx^{2}} \right)^{2} \]
Applying the exponent rule \((a^m)^n = a^{mn}\):
\[ \left[ 1 + \left(\frac{dy}{dx}\right)^{2} \right]^{3} = \left( \frac{d^{2}y}{dx^{2}} \right)^{2} \]
Now, the equation is in polynomial form. The highest order derivative is \(\frac{d^{2}y}{dx^{2}}\). The power (exponent) to which it is raised is \(2\).
Therefore, Degree = 2.
Step 3: Final Answer:
After squaring both sides to remove fractional exponents, the second-order derivative is raised to the second power, giving a degree of \(2\).