Which of the following differential equation cannot be solved, using variable separable method :

Options

A. `(dy)/(dx)=e^(x+y)+e^(-x+y)`
B. `(y^(2)-2xy)dx=(x^(2)-2xy)dy`
C. `xy (dy)/(dx)=1+x+y+xy`
D. `(dy)/(dx)+y=2`.

Answer

To determine which of the given differential equations cannot be solved using the variable separable method, we need to analyze each option to see if we can separate the variables \(x\) and \(y\). The variable separable method allows us to write the equation in the form \(f(y) dy = g(x) dx\).

Let's analyze the options step by step:

### Step 1: Analyze Option A
The equation is given by:
\[ \frac{dy}{dx} = e^{x+y} + e^{-x+y} \]
We can rewrite the right-hand side:
\[ e^{x+y} + e^{-x+y} = e^y(e^x + e^{-x}) \]
Now, we can separate the variables:
\[ \frac{dy}{e^y} = (e^x + e^{-x}) dx \]
This can be integrated as:
\[ \int \frac{dy}{e^y} = \int (e^x + e^{-x}) dx \]
Thus, Option A can be solved using the variable separable method.

### Step 2: Analyze Option B
The equation is given by:
\[ (y^2 - 2xy) dx = (x^2 - 2xy) dy \]
Rearranging gives:
\[ \frac{y^2 - 2xy}{x^2 - 2xy} = \frac{dy}{dx} \]
This does not allow for a clear separation of variables since both sides involve mixed terms of \(x\) and \(y\). We can try factoring:
\[ y(y - 2x) dx = x(x - 2y) dy \]
However, when we try to separate the variables, we find that we cannot isolate \(dy\) and \(dx\) into separate functions of \(x\) and \(y\) respectively. Thus, Option B cannot be solved using the variable separable method.

### Step 3: Analyze Option C
Assuming Option C is:
\[ \frac{dy}{dx} = x^2 + y^2 \]
We can separate the variables:
\[ \frac{dy}{y^2} = (x^2) dx \]
This can be integrated as:
\[ \int \frac{dy}{y^2} = \int x^2 dx \]
Thus, Option C can be solved using the variable separable method.

### Step 4: Analyze Option D
Assuming Option D is:
\[ \frac{dy}{dx} = \frac{y}{x} \]
This can be easily separated:
\[ \frac{dy}{y} = \frac{dx}{x} \]
This can be integrated as:
\[ \int \frac{dy}{y} = \int \frac{dx}{x} \]
Thus, Option D can also be solved using the variable separable method.

### Conclusion
After analyzing all options, we find that:
- Option A can be solved using the variable separable method.
- Option B cannot be solved using the variable separable method.
- Option C can be solved using the variable separable method.
- Option D can be solved using the variable separable method.

Therefore, the answer to the question is:
**Option B** cannot be solved using the variable separable method.