The order and degree of the differential equation `sqrt((d^(2)y)/(dx^2))= root(3)((dy)/(dx)+5)` are respectively

Options

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

Answer

To determine the order and degree of the given differential equation

\[ \sqrt{\frac{d^2y}{dx^2}} = \sqrt[3]{\frac{dy}{dx} + 5} \]

we will follow these steps:

### Step 1: Eliminate the square root and cube root
To eliminate the square root and cube root, we will raise both sides to appropriate powers. The left side has a square root, which we can eliminate by squaring both sides, and the right side has a cube root, which we can eliminate by cubing both sides.

1. Square both sides:
\[ \left(\sqrt{\frac{d^2y}{dx^2}}\right)^2 = \left(\sqrt[3]{\frac{dy}{dx} + 5}\right)^2 \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = \left(\frac{dy}{dx} + 5\right)^{\frac{2}{3}} \]

2. Now, cube both sides to eliminate the cube root:
\[ \left(\frac{d^2y}{dx^2}\right)^3 = \left(\left(\frac{dy}{dx} + 5\right)^{\frac{2}{3}}\right)^3 \]
This simplifies to:
\[ \left(\frac{d^2y}{dx^2}\right)^3 = \left(\frac{dy}{dx} + 5\right)^2 \]

### Step 2: Identify the order and degree
Now we can analyze the resulting equation:

\[ \left(\frac{d^2y}{dx^2}\right)^3 = \left(\frac{dy}{dx} + 5\right)^2 \]

- The **order** of a differential equation is the highest derivative present. Here, the highest derivative is \(\frac{d^2y}{dx^2}\), which is of order 2.
- The **degree** of a differential equation is the power of the highest derivative when the equation is a polynomial in derivatives. In this case, the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3. Therefore, the degree is 3.

### Final Result
Thus, the order and degree of the given differential equation are:
- **Order**: 2
- **Degree**: 3