The order and degree of `(1 + ((dy)/(dx)) ^(3) ) ^((2)/(3)) = 8(d ^(3)y)/( dx ^(3))` are respectively

Options

A. `3,1`
B. `1,3`
C. `3,3`
D. `1,1`

Answer

To determine the order and degree of the given differential equation:

\[ (1 + \left(\frac{dy}{dx}\right)^3)^{\frac{2}{3}} = 8\frac{d^3y}{dx^3} \]

we will follow these steps:

### Step 1: Eliminate the fractional exponent

To eliminate the fractional exponent, we will raise both sides of the equation to the power of 3.

\[ \left((1 + \left(\frac{dy}{dx}\right)^3)^{\frac{2}{3}}\right)^3 = \left(8\frac{d^3y}{dx^3}\right)^3 \]

This simplifies to:

\[ (1 + \left(\frac{dy}{dx}\right)^3)^2 = 512\left(\frac{d^3y}{dx^3}\right)^3 \]

### Step 2: Identify the highest order derivative

Now we need to identify the highest order derivative present in the equation.

- The left side contains \(\frac{dy}{dx}\), which is a first-order derivative.
- The right side contains \(\frac{d^3y}{dx^3}\), which is a third-order derivative.

The highest order derivative is \(\frac{d^3y}{dx^3}\), which is of order 3.

### Step 3: Determine the degree of the differential equation

The degree of a differential equation is determined by the highest power of the highest order derivative when the equation is expressed in a polynomial form.

In our case, the highest order derivative is \(\frac{d^3y}{dx^3}\) and it appears with a power of 3 in the equation:

\[ (1 + \left(\frac{dy}{dx}\right)^3)^2 = 512\left(\frac{d^3y}{dx^3}\right)^3 \]

Thus, the degree of the differential equation is 3.

### Final Answer

The order of the differential equation is 3, and the degree is also 3.