The given differential equation is:
$\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)^3 = 0$
Step 1: Expand the derivative
$\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)^3 = 3\left(\frac{dy}{dx}\right)^2 \frac{d^2y}{dx^2} = 0$
Step 2: Identify order and degree
Step 3: Compute $(p - q)$
$(p - q) = 2 - 1 = 1$
∴ Final Answer: $(p - q) = 1$
The given differential equation is \(\frac{d}{dx}(\frac{dy}{dx})^{3}=0\).
First, we simplify the equation:
\(\frac{d}{dx}(\frac{dy}{dx})^{3} = \frac{d}{dx}((\frac{dy}{dx})^{3}) = 0\)
Differentiating \(\left(\frac{dy}{dx}\right)^3\) with respect to \(x\) gives:
\(3\left(\frac{dy}{dx}\right)^2 \cdot \frac{d^2y}{dx^2} = 0\)
The order of a differential equation is the highest order derivative present in the equation. Here, the highest order derivative is \(\frac{d^2y}{dx^2}\), which is the second derivative. So, the order \(p = 2\).
The degree of a differential equation is the power of the highest order derivative, when the differential equation is expressed in a form where all derivatives are free from radicals and fractions. In this case, the power of \(\frac{d^2y}{dx^2}\) is 1. So, the degree \(q = 1\).
Therefore, \(p - q = 2 - 1 = 1\).
Correct Answer: 1
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