The order and the degree of the differential equation \( \left(\displaystyle\frac{d^{4}y}{dx^{4}}\right)^{2}=\left[x+\left(\displaystyle\frac{dy}{dx}\right)^{2}\right]^{3} \) are
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → order 4 , degree 2
The highest derivative present is \( \frac{d^{4}y}{dx^{4}} \), so the order is 4.
The right side expands into a polynomial in \( \frac{dy}{dx} \), so the equation is a polynomial in its derivatives. The highest derivative is squared, so the degree is 2.
Why the other options are incorrect
Option (1) order 1;degree 6 — Incorrect. The order is 4, not 1. The 6 is a power of \( \frac{dy}{dx} \) after expansion, not of the highest derivative.
Option (3) order 1, degree 2 — Incorrect. The order is 4, not 1.
Option (4) order 4,degree not defined — Incorrect. The degree is defined, since the equation is a polynomial in its derivatives.