Question

Find the order and degree of the differential equation: \[ \sqrt{1+\left(\frac{dy}{dx}\right)^2}=\frac{d^2y}{dx^2} \]

• A. \( \text{Order }2,\ \text{Degree }2 \)
• B. \( \text{Order }2,\ \text{Degree }1 \)
• C. \( \text{Order }1,\ \text{Degree }2 \)
• D. \( \text{Order }2,\ \text{Degree }\frac{1}{2} \)

Hint:

Whenever radicals or fractional powers involve derivatives, first remove them before finding the degree of the differential equation.

Answer 1

The Correct Option is A

Concept: The order of a differential equation is the order of the highest derivative present in the equation. The degree of a differential equation is the power of the highest-order derivative after the equation has been made free from radicals and fractional powers involving derivatives. Before determining the degree, the equation must be converted into polynomial form with respect to derivatives.

Step 1: Write the given differential equation The given differential equation is \[ \sqrt{1+\left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2} \] We observe that the highest derivative present is \[ \frac{d^2y}{dx^2} \] Therefore, the order is expected to be 2.

Step 2: Remove the radical sign To determine the degree, we must first eliminate the square root. Squaring both sides gives \[ \left( \sqrt{1+\left(\frac{dy}{dx}\right)^2} \right)^2 = \left( \frac{d^2y}{dx^2} \right)^2 \] Hence, \[ 1+\left(\frac{dy}{dx}\right)^2 = \left(\frac{d^2y}{dx^2}\right)^2 \] This equation is now polynomial in derivatives.

Step 3: Determine the order The highest-order derivative appearing in the equation is \[ \frac{d^2y}{dx^2} \] Therefore, \[ \boxed{\text{Order}=2} \]

Step 4: Determine the degree The highest-order derivative is \[ \frac{d^2y}{dx^2} \] and its exponent is \[ 2 \] Therefore, \[ \boxed{\text{Degree}=2} \] Final Answer: \[ \boxed{\text{Order }2,\ \text{Degree }2} \]