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The order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) respectively are:
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Q: The order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) respectively are:

Question Analysis & Solution

Step-by-Step Solution

  1. First, rewrite the given differential equation to eliminate the radical: \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) Taking the power of \(\frac{1}{3}\) on both sides is not needed as we are looking for order and degree. Instead, we raise both sides to the power of 1.
  2. The order of a differential equation is the highest order derivative present in the equation. In this case, the highest order derivative is \(\frac{d^{2}y}{dx^{2}}\), which is the second derivative. Therefore, the order is 2.
  3. The degree of a differential equation is the power of the highest order derivative, after the equation has been expressed in a form where all derivatives are free from radicals and fractions. The given equation is \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\). The power of the highest order derivative \(\frac{d^{2}y}{dx^{2}}\) is 1. Therefore, the degree is 1.
  4. Thus, the order and degree of the given differential equation are 2 and 1, respectively.

Correct Answer: (C) 2, 1

AI Suggestion: Option C
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