The differential equation of ` tan(x-y) = cx ` is

Options

A. `2x (1- (dy)/(dx)) = sin 2(x-y)`
B. `2y(1- (dy)/(dx))= cos 2 (x-y)`
C. `x(1- (dy)/(dx))= cos 2(x-y) `
D. `y(1- (dy)/(dx)) = sin 2(x-y)`

Answer

To find the differential equation of the given expression \( \tan(x - y) = cx \), we will differentiate both sides with respect to \( x \).

### Step-by-step Solution:

1. **Differentiate both sides**:
We start with the equation:
\[ \tan(x - y) = cx \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}[\tan(x - y)] = \frac{d}{dx}[cx] \]

2. **Apply the chain rule on the left side**:
Using the chain rule, the derivative of \( \tan(u) \) where \( u = x - y \) gives:
\[ \sec^2(x - y) \cdot \frac{d}{dx}(x - y) = c \]
Now, differentiate \( x - y \):
\[ \frac{d}{dx}(x - y) = 1 - \frac{dy}{dx} \]
Thus, we can rewrite the equation as:
\[ \sec^2(x - y)(1 - \frac{dy}{dx}) = c \]

3. **Substitute \( c \)**:
From the original equation \( \tan(x - y) = cx \), we can express \( c \) as:
\[ c = \frac{\tan(x - y)}{x} \]
Substituting this back into the differentiated equation gives:
\[ \sec^2(x - y)(1 - \frac{dy}{dx}) = \frac{\tan(x - y)}{x} \]

4. **Multiply both sides by \( x \)**:
To eliminate the fraction, we multiply both sides by \( x \):
\[ x \sec^2(x - y)(1 - \frac{dy}{dx}) = \tan(x - y) \]

5. **Final form**:
Rearranging gives us the differential equation:
\[ x \sec^2(x - y) - x \sec^2(x - y) \frac{dy}{dx} = \tan(x - y) \]

### Final Differential Equation:
The final differential equation derived from \( \tan(x - y) = cx \) is:
\[ x \sec^2(x - y) - x \sec^2(x - y) \frac{dy}{dx} = \tan(x - y) \]