Discussion
The order and degree of the differential equation \(\sqrt{\dfrac{d^2y}{dx^2}\sqrt{\dfrac{dy}{dx}}} = 3\) is:
Answer: C
Remove the radicals by squaring, taking the derivatives as the variables.
\[ \frac{d^2y}{dx^2}\sqrt{\frac{dy}{dx}} = 9 \]
\[ \left(\frac{d^2y}{dx^2}\right)^2 \frac{dy}{dx} = 81 \]
The highest derivative is \(d^2y/dx^2\), so the order is 2, and it appears to the power 2 once the radicals are gone, so the degree is 2. The answer is \(2, 2\), option C. The degree is defined only after the equation is a polynomial in its derivatives, which is why the radicals have to be cleared first.
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