`"Solve the following differential equations"` :
`(dy)/(dx)=cot^(2)(x+y)`.

Answer

To solve the differential equation \(\frac{dy}{dx} = \cot^2(x + y)\), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ \frac{dy}{dx} = \cot^2(x + y) \]
This equation cannot be separated directly, so we will use a substitution to simplify it.

### Step 2: Use Substitution
Let \(v = x + y\). Then, we have:
\[ y = v - x \quad \text{and thus} \quad \frac{dy}{dx} = \frac{dv}{dx} - 1 \]
Substituting this into the original equation gives:
\[ \frac{dv}{dx} - 1 = \cot^2(v) \]
Rearranging this, we get:
\[ \frac{dv}{dx} = \cot^2(v) + 1 \]

### Step 3: Simplify Using Trigonometric Identity
Using the identity \(1 + \cot^2(v) = \csc^2(v)\), we can rewrite the equation as:
\[ \frac{dv}{dx} = \csc^2(v) \]

### Step 4: Separate Variables
Now we can separate the variables:
\[ \csc^2(v) \, dv = dx \]

### Step 5: Integrate Both Sides
Next, we integrate both sides:
\[ \int \csc^2(v) \, dv = \int dx \]
The integral of \(\csc^2(v)\) is \(-\cot(v)\), so we have:
\[ -\cot(v) = x + C \]
where \(C\) is the constant of integration.

### Step 6: Substitute Back for \(v\)
Now we substitute back \(v = x + y\):
\[ -\cot(x + y) = x + C \]
This can be rearranged to give:
\[ \cot(x + y) = -x - C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \cot(x + y) = -x - C \]