The order and degree of the differential equation \(\left[\left(\displaystyle\frac{d^2y}{dx^2}\right)^2-3\right]^{\frac{1}{3}}=2\left(\displaystyle\frac{dy}{dx}\right)^{\frac{1}{4}}\) are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → order = 2, degree = 8
The degree is read only after the equation is made a polynomial in the derivatives. The fractional powers are \(\frac{1}{3}\) and \(\frac{1}{4}\), so raise both sides to the power 12 (the LCM of 3 and 4):
\( \displaystyle \left[\left(\frac{d^2y}{dx^2}\right)^2-3\right]^{4} = 2^{12}\left(\frac{dy}{dx}\right)^{3} \)
The highest derivative is \(\frac{d^2y}{dx^2}\), so the order is 2.
Expanding the left side, the highest power of \(\frac{d^2y}{dx^2}\) is
\( \displaystyle \left[\left(\frac{d^2y}{dx^2}\right)^2\right]^{4} = \left(\frac{d^2y}{dx^2}\right)^{8} \)
So the degree is 8.
Tip: Reading the degree before clearing the fractional powers gives 2, Option (1). Remove radicals and fractional powers first, then take the power of the highest derivative.