The order and the degree of the differential equation \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^3 + x\left(\displaystyle\frac{dy}{dx}\right)^5 - y = e^x\) are respectively
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 and 3
The order is the order of the highest derivative present. The highest derivative here is \(\frac{d^2y}{dx^2}\), so the order is 2.
The equation is a polynomial in the derivatives, so its degree is the highest power of that highest derivative. \(\frac{d^2y}{dx^2}\) appears with power 3, so the degree is 3.
Why the other options are incorrect
Option (1) 2 and 5 — Incorrect. 5 is the power of \(\frac{dy}{dx}\), which is not the highest derivative; the degree is read from \(\frac{d^2y}{dx^2}\) only.
Option (2) 1 and 5 — Incorrect. This reads both values from the \(\frac{dy}{dx}\) term; the second derivative is present, so the order is 2.
Option (4) 3 and 2 — Incorrect. The two values have been swapped; the order is 2 and the degree is 3.