Question

The order and degree of the differential equation \(\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}=\left(\frac{d^{2}y}{dx^{2}}\right)^{\frac{1}{3}}\) are

• A. order =2, degree = 2
• B. order =2, degree=\(\frac{1}{3}\)
• C. order = 3, degree is not defined
• D. order = 2, degree = 3

Hint:

Clear the fractional powers by raising both sides to the power 6, then read the power of the highest derivative.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The order is the highest derivative present. The degree is the power of that highest derivative after the equation is made free of radicals and fractional powers in the derivatives.

Step 2: Find the order.
The highest derivative is \(\frac{d^{2}y}{dx^{2}}\). So the order is 2.

Step 3: Remove the radical and fractional power.
Square both sides: \(1+\left(\frac{dy}{dx}\right)^{2}=\left(\frac{d^{2}y}{dx^{2}}\right)^{\frac{2}{3}}\).
Now cube both sides:
\[ \left[1+\left(\frac{dy}{dx}\right)^{2}\right]^{3}=\left(\frac{d^{2}y}{dx^{2}}\right)^{2} \]

Step 4: Find the degree.
Now the equation is a polynomial in the derivatives. The highest derivative \(\frac{d^{2}y}{dx^{2}}\) has power 2. So the degree is 2.

Step 5: Check the other options.
Degree cannot be \(\frac{1}{3}\), because degree is defined only after removing fractional powers. Order 3 is wrong since no third derivative exists. Degree 3 is wrong because after clearing powers the second derivative appears squared, not cubed.

Final Answer:
Order = 2 and degree = 2, option 1. \[ \boxed{\text{order}=2,\ \text{degree}=2} \]