Represent the following families of curves by forming the corresponding differential (a, b : parameters) :
`y=ax`

Answer

To represent the family of curves given by the equation \( y = ax \) in terms of a differential equation, we will follow these steps:

### Step 1: Differentiate the equation
We start with the equation:
\[ y = ax \]
We differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = a \]

### Step 2: Express \( a \) in terms of \( y \) and \( x \)
From the differentiated equation, we can express \( a \) as:
\[ a = \frac{dy}{dx} \]

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

### Step 4: Rearrange the equation
Rearranging gives:
\[ \frac{dy}{dx} = \frac{y}{x} \]

### Step 5: Write the differential equation in standard form
We can rewrite this in standard form:
\[ \frac{dy}{dx} - \frac{y}{x} = 0 \]

### Final Result
Thus, the corresponding differential equation representing the family of curves \( y = ax \) is:
\[ \frac{dy}{dx} - \frac{y}{x} = 0 \]

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