Write the order of the differential equation representing the family of curves `y= kx +k^(4)`.

Answer

To determine the order of the differential equation representing the family of curves given by \( y = kx + k^4 \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the family of curves:
\[ y = kx + k^4 \]
We differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = k \]

### Step 2: Express \( k \) in terms of \( \frac{dy}{dx} \)
From the differentiation, we have:
\[ k = \frac{dy}{dx} \]

### Step 3: Substitute \( k \) back into the original equation
Now, we substitute \( k \) back into the original equation:
\[ y = \left(\frac{dy}{dx}\right)x + \left(\frac{dy}{dx}\right)^4 \]

### Step 4: Rearrange the equation
This gives us the differential equation:
\[ y = \frac{dy}{dx} \cdot x + \left(\frac{dy}{dx}\right)^4 \]

### Step 5: Identify the order of the differential equation
The highest derivative present in the equation is \( \frac{dy}{dx} \), which is a first derivative. Since there are no higher derivatives (like \( \frac{d^2y}{dx^2} \)), we conclude that the order of the differential equation is:
\[ \text{Order} = 1 \]

### Final Answer
The order of the differential equation representing the family of curves \( y = kx + k^4 \) is **1**.

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