To determine the degree of the given differential equation:
[tex]\[ \left(\frac{d^2 y}{d x^2}\right)^3 + \frac{d y}{d x} - y = 0, \][/tex]
we need to follow these steps:
1. Identify the highest order derivative:
- The differential equation contains derivatives of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]. The highest order derivative present in this equation is the second derivative, [tex]\( \frac{d^2 y}{d x^2} \)[/tex].
2. Determine the power of the highest order derivative:
- In the equation, the highest order derivative, [tex]\( \frac{d^2 y}{d x^2} \)[/tex], is raised to a power. Here, [tex]\( \left( \frac{d^2 y}{d x^2} \right)^3 \)[/tex] indicates that the highest order derivative is raised to the power of 3.
3. Conclusion:
- The degree of a differential equation is defined as the highest power to which the highest order derivative is raised.
- In this case, the highest order derivative [tex]\( \frac{d^2 y}{d x^2} \)[/tex] is raised to the power of 3.
Thus, the degree of the given differential equation is [tex]\(\boxed{3}\)[/tex].
