Question

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

• A. order = 2, degree = 1
• B. order = 2, degree = 2
• C. order = 1, degree = 2
• D. order = 1, degree = 1

Answer 1

The Correct Option is B

To determine the order and degree of the given differential equation:
\(\frac{d^2y}{dx^2}=(1+\frac{dy}{dx})^{\frac{1}{2}}\), we need to follow these steps:

1. Order of the differential equation:
   The order is defined as the highest derivative present in the equation. Here, the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative. Therefore, the order of the differential equation is 2.
2. Degree of the differential equation:
   The degree is determined by the power to which the highest order derivative is raised, but only after the equation is made polynomial in derivatives. The given equation is:
   \(\frac{d^2y}{dx^2}=(1+\frac{dy}{dx})^{\frac{1}{2}}\).
   This equation is not a polynomial in derivatives due to the fractional power \(\frac{1}{2}\). To find the degree, we first eliminate the radical by squaring both sides:
   \(\left(\frac{d^2y}{dx^2}\right)^2=(1+\frac{dy}{dx})\).
   Now the equation is polynomial in derivatives. The highest derivative \((\frac{d^2y}{dx^2})\) is raised to the power of 2, determining the degree as 2.

As a result, the differential equation's order is 2 and its degree is 2. Therefore, the correct answer is:

Order = 2, Degree = 2