The degree and order of the differential equation \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^{4/5}=10\displaystyle\frac{dy}{dx}+2\) are:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Degree 4, Order 2
The order is the order of the highest derivative in the equation. The degree is the power of that highest derivative once the equation is a polynomial in the derivatives (no fractional powers).
The highest derivative is \(\frac{d^2y}{dx^2}\), so the order is 2.
Raising both sides to the power 5 removes the fractional power:
\( \displaystyle \left(\frac{d^2y}{dx^2}\right)^{4} = \left(10\frac{dy}{dx}+2\right)^{5} \)
Now the highest derivative \(\frac{d^2y}{dx^2}\) appears with power 4, so the degree is 4.
Why the other options are incorrect
Option (1) Degree 2, Order 5 — Incorrect. No fifth derivative appears in the equation, so the order cannot be 5.
Option (2) Degree 5, Order 1 — Incorrect. The power 5 belongs to \(\frac{dy}{dx}\), a first derivative, which is not the highest derivative present.
Option (3) Degree 20, Order 2 — Incorrect. 20 = 4 × 5 multiplies the two powers; the degree is the power of the highest derivative alone, which is 4.