The differential equation which represents the family of curves given by tan y = `c (1 - e^(x))` is

Options

A. `e^(x) tan y dx + (1 - e^(x)) dy = 0`
B. `e^(x) tan y dx + (1 - e^(x)) sec^(2) y dy = 0`
C. `e^(x) (1- e^(x)) dx + tan y dy = 0`
D. `e^(x) tan y dy + (1 - e^(x)) dx = 0`

Answer

To find the differential equation that represents the family of curves given by the equation \( \tan y = c(1 - e^x) \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ \tan y = c(1 - e^x) \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\tan y) = \frac{d}{dx}(c(1 - e^x)) \]
Using the chain rule on the left side, we have:
\[ \sec^2 y \frac{dy}{dx} = c \cdot 0 - c e^x \]
This simplifies to:
\[ \sec^2 y \frac{dy}{dx} = -c e^x \]

### Step 2: Solve for \( c \)
From the original equation \( \tan y = c(1 - e^x) \), we can express \( c \) in terms of \( y \) and \( x \):
\[ c = \frac{\tan y}{1 - e^x} \]

### Step 3: Substitute \( c \) back into the differentiated equation
Now, substitute \( c \) into the differentiated equation:
\[ \sec^2 y \frac{dy}{dx} = -\left(\frac{\tan y}{1 - e^x}\right) e^x \]
This gives us:
\[ \sec^2 y \frac{dy}{dx} = -\frac{e^x \tan y}{1 - e^x} \]

### Step 4: Rearranging the equation
We can rearrange this equation to isolate terms:
\[ (1 - e^x) \sec^2 y \frac{dy}{dx} + e^x \tan y = 0 \]

### Final Form of the Differential Equation
Thus, the differential equation that represents the family of curves is:
\[ (1 - e^x) \sec^2 y \frac{dy}{dx} + e^x \tan y = 0 \]