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

Answer

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

### Step 1: Start with the given equation
The given equation is:
\[ y = e^{ax} \]

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm of both sides, we have:
\[ \ln y = ax \]

### Step 3: Express \( a \) in terms of \( y \) and \( x \)
From the equation \( \ln y = ax \), we can express \( a \) as:
\[ a = \frac{\ln y}{x} \]

### Step 4: Differentiate \( y \) with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{ax}) = e^{ax} \cdot \frac{d}{dx}(ax) = e^{ax} \cdot a \]
Substituting \( a \) from the previous step:
\[ \frac{dy}{dx} = e^{ax} \cdot \frac{\ln y}{x} \]

### Step 5: Substitute \( e^{ax} \) with \( y \)
Since \( e^{ax} = y \), we can replace \( e^{ax} \) in the equation:
\[ \frac{dy}{dx} = y \cdot \frac{\ln y}{x} \]

### Step 6: Rearranging the equation
To express this in a standard form, we can multiply both sides by \( x \):
\[ x \frac{dy}{dx} = y \ln y \]

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

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