To determine the degree of a differential equation, first remove any radicals or fractional exponents by raising both sides of the equation to the appropriate powers. This ensures the equation is polynomial in terms of its derivatives. Always focus on the highest order derivative and its highest power once the equation is free from any fractional terms. This process helps in clearly identifying the degree.
The provided differential equation is:
\[ \left( 1 - \left( \frac{dy}{dx} \right)^2 \right)^{3/2} = k \frac{d^2y}{dx^2}. \]
The degree of a differential equation is defined as the highest power of the highest order derivative after all fractional powers and radicals involving derivatives have been eliminated.
To remove the fractional exponent, raise both sides of the equation to the power of \(\frac{2}{3}\):
\[ 1 - \left( \frac{dy}{dx} \right)^2 = \left( k \frac{d^2y}{dx^2} \right)^{2/3}. \]
To render the equation polynomial in terms of its derivatives, raise both sides to the power of 3:
\[ \left( 1 - \left( \frac{dy}{dx} \right)^2 \right)^3 = \left( k \frac{d^2y}{dx^2} \right)^2. \]
In this transformed equation, the highest order derivative present is \(\frac{d^2y}{dx^2}\), and its maximum power is 2.
Consequently, the degree of this differential equation is 2.
Determine the degree of the following differential equation: \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}} = \frac{d^2y}{dx^2} \]