Question

The order and degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = \frac{d^2y}{dx^2}$ are respectively:

• A. $2$ and $2$
• B. $2$ and $3$
• C. $1$ and $2$
• D. $2$ and $1$

Hint:

Always clear fractional indices first! The degree of a differential equation can only be defined when it is written as a polynomial in its derivatives.

Answer 1

The Correct Option is A

Step 1: Recall order and degree.
The order is the highest derivative that appears. The degree is the power of that highest derivative, but only after the equation is free of roots and fractional powers. So we may need to clean the equation first.

Step 2: Look at the equation.
The equation is $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = \frac{d^2y}{dx^2}$. There is a fractional power $\frac{3}{2}$ on the left, so degree cannot be read yet.

Step 3: Remove the fractional power.
To clear the $\frac{3}{2}$, square both sides. Squaring doubles each exponent.
\[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = \left(\frac{d^2y}{dx^2}\right)^2 \]

Step 4: Find the highest derivative.
The highest derivative present is $\frac{d^2y}{dx^2}$, the second derivative. So the order is $2$.

Step 5: Find the degree.
After squaring, the highest derivative $\frac{d^2y}{dx^2}$ appears to the power $2$. So the degree is $2$.

Step 6: State both.
Order is $2$ and degree is $2$.
\[ \boxed{\text{order } 2, \text{ degree } 2} \]