Discussion

The order and degree of the differential equation

\[ \begin{aligned} &\left[1 + 3\frac{dy}{dx}\right]^{\frac{2}{3}} \\ &= 4\frac{d^{3}y}{dx^{3}} \end{aligned} \]

is :

Answer: C

The order and degree are (3, 3).

The highest derivative is \(d^{3}y/dx^{3}\), so the order is 3. To find the degree, first clear the fractional power by cubing both sides:

\[ \begin{aligned} &\left(1 + 3\frac{dy}{dx}\right)^{2} \\ &= 64\left(\frac{d^{3}y}{dx^{3}}\right)^{3} \end{aligned} \]

The equation is now a polynomial in the derivatives, and the highest derivative appears to the power 3, so the degree is 3.

(3, 2) reads the degree from the exponent 2 on the left, which belongs to the first derivative, and the other options miscount the order or degree.

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