The degree and order of the differential equation `y=px+root3(a^(2)p^(2)+b^(2)),` where `p = (dy)/(dx) ` are respectively

Options

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

Explanation

To find the degree and order of the given differential equation \( y = px + \sqrt[3]{a^2 p^2 + b^2} \), where \( p = \frac{dy}{dx} \), we can follow these steps:

### Step 1: Substitute \( p \) with \( \frac{dy}{dx} \)

We start by substituting \( p \) in the equation:

\[ y = \left(\frac{dy}{dx}\right)x + \sqrt[3]{a^2 \left(\frac{dy}{dx}\right)^2 + b^2} \]

### Step 2: Rearrange the equation

Next, we rearrange the equation to isolate terms involving \( \frac{dy}{dx} \):

\[ y - x \frac{dy}{dx} = \sqrt[3]{a^2 \left(\frac{dy}{dx}\right)^2 + b^2} \]

### Step 3: Cube both sides to eliminate the cube root

To eliminate the cube root, we cube both sides of the equation:

\[ \left(y - x \frac{dy}{dx}\right)^3 = a^2 \left(\frac{dy}{dx}\right)^2 + b^2 \]

### Step 4: Expand the left-hand side

Using the binomial expansion, we expand the left-hand side:

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

### Step 5: Set the equation to zero

Now we can set the equation to zero:

\[ y^3 - 3y^2\left(x \frac{dy}{dx}\right) + 3y\left(x \frac{dy}{dx}\right)^2 - (x \frac{dy}{dx})^3 - a^2 \left(\frac{dy}{dx}\right)^2 - b^2 = 0 \]

### Step 6: Identify the order and degree

In the final equation, the highest derivative is \( \frac{dy}{dx} \), which appears in the form \( (x \frac{dy}{dx})^3 \) and \( (x \frac{dy}{dx})^2 \).

- **Order**: The order of the differential equation is the highest derivative present, which is 1 (since we have \( \frac{dy}{dx} \)).
- **Degree**: The degree is determined by the highest power of the highest derivative. Here, the highest power of \( \frac{dy}{dx} \) in the equation is 3 (from \( (x \frac{dy}{dx})^3 \)).

Thus, the order is 1 and the degree is 3.

### Final Answer:
- **Order**: 1
- **Degree**: 3