The order and degree of the differential equation `(d^(3)y)/(dx^(3))=root5(1+((dy)/(dx)))` is :

Options

A. 3, 7
B. 3, 1
C. 3, 5
D. 1, 3

Answer

To determine the order and degree of the given differential equation

\[ \frac{d^3y}{dx^3} = \sqrt[5]{1 + \frac{dy}{dx}}, \]

we will follow these steps:

### Step 1: Identify the highest derivative
The left side of the equation contains the term \(\frac{d^3y}{dx^3}\), which is the third derivative of \(y\). This indicates that the highest order derivative in the equation is the third derivative.

**Hint:** The order of a differential equation is determined by the highest derivative present in the equation.

### Step 2: Determine the order
Since the highest derivative is \(\frac{d^3y}{dx^3}\), the order of the differential equation is 3.

**Hint:** The order is simply the number of times the function \(y\) is differentiated.

### Step 3: Rewrite the equation to find the degree
Next, we need to express the equation in a form where we can easily identify the degree. The right side of the equation involves a root, which we can eliminate by raising both sides to the power of 5:

\[ \left(\frac{d^3y}{dx^3}\right)^5 = 1 + \frac{dy}{dx}. \]

**Hint:** To find the degree, convert any roots or fractional powers to whole numbers by raising the equation to an appropriate power.

### Step 4: Identify the highest order derivative and its power
Now, the equation is in a polynomial form. The highest order derivative is still \(\frac{d^3y}{dx^3}\), and it is raised to the power of 5.

**Hint:** The degree of a differential equation is the exponent of the highest order derivative when the equation is expressed as a polynomial in derivatives.

### Step 5: Determine the degree
Thus, the degree of the differential equation is 5.

### Final Answer
The order and degree of the differential equation are:

\[ \text{Order} = 3, \quad \text{Degree} = 5. \]

So, the final answer is \(3, 5\).

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