Question

The order and degree of the differential equation $$\frac{d^2y}{dx^2} = \sqrt{\frac{dy}{dx}}$$ are respectively

• A. 2, 3
• B. 3, 3
• C. 2, 2
• D. 1, 3

Hint:

Never look at the degree until you've completely cleared out fractional exponents and roots surrounding your derivative terms! Squaring or cubing early prevents falling into standard trap options.

Answer 1

The Correct Option is C

Step 1: Recall the two definitions.
The order is the highest derivative that appears; the degree is the power of that highest derivative once the equation is free of radicals and fractional powers.
Step 2: Read the given equation.
$\dfrac{d^2y}{dx^2} = \sqrt{\dfrac{dy}{dx}}$. The highest derivative present is $\dfrac{d^2y}{dx^2}$.
Step 3: Note the order immediately.
Since the second derivative is the highest, the order is $2$.
Step 4: Remove the radical to judge the degree.
A square root means a fractional power, so we square both sides: $\left(\dfrac{d^2y}{dx^2}\right)^2 = \dfrac{dy}{dx}$.
Step 5: Read off the degree.
Now the highest derivative $\dfrac{d^2y}{dx^2}$ appears to the power $2$, so the degree is $2$.
Step 6: State the pair.
Order $= 2$, degree $= 2$.
\[ \boxed{\text{Order} = 2,\ \text{Degree} = 2} \]