The differential equation which represents the family of plane curves `y=exp(cx)`, is

Options

A. `y'=cy`
B. `xy'-logy=0`
C. `xlogy=yy'`
D. `y log y=xy'`

Answer

To find the differential equation that represents the family of plane curves given by \( y = e^{cx} \), we will follow these steps:

### Step 1: Write the given equation
The equation of the family of curves is:
\[ y = e^{cx} \]

### Step 2: Differentiate the equation
We differentiate \( y \) with respect to \( x \). Using the chain rule, we get:
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{cx}) = e^{cx} \cdot \frac{d}{dx}(cx) = e^{cx} \cdot c \]
Thus, we can write:
\[ \frac{dy}{dx} = c e^{cx} \]

### Step 3: Substitute \( e^{cx} \) with \( y \)
Since \( y = e^{cx} \), we can substitute \( e^{cx} \) in our derivative equation:
\[ \frac{dy}{dx} = c y \]

### Step 4: Eliminate the constant \( c \)
To express the equation in terms of \( y \) and its derivative without the constant \( c \), we can rearrange the equation:
\[ \frac{dy}{dx} - \frac{y}{x} = 0 \]

### Final Differential Equation
The final differential equation that represents the family of curves \( y = e^{cx} \) is:
\[ \frac{dy}{dx} - \frac{y}{x} = 0 \]