Solve `(dy)/(dx)=cos(x+y)-sin(x+y)`.

Explanation

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

### Step 1: Substitute \(v = x + y\)
Let \(v = x + y\). Then, we can express \(y\) in terms of \(v\) and \(x\):
\[ y = v - x \]
Now, differentiate both sides with respect to \(x\):
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 2: Rewrite the differential equation
Substituting \(\frac{dy}{dx}\) into the original differential equation gives:
\[ \frac{dv}{dx} - 1 = \cos(v) - \sin(v) \]
Rearranging this, we have:
\[ \frac{dv}{dx} = \cos(v) - \sin(v) + 1 \]

### Step 3: Separate variables
Now, we can separate the variables:
\[ \frac{dv}{\cos(v) - \sin(v) + 1} = dx \]

### Step 4: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{dv}{\cos(v) - \sin(v) + 1} = \int dx \]
The right side integrates to:
\[ x + C \]
where \(C\) is the constant of integration.

### Step 5: Simplify the left-hand side
To simplify the left-hand side, we can use trigonometric identities. We can rewrite \(\cos(v) - \sin(v) + 1\) using half-angle identities:
\[ \cos(v) - \sin(v) + 1 = 1 + \cos(v) - \sin(v) \]
This can be tricky to integrate directly, so we will use a substitution if necessary.

### Step 6: Solve the integral
Let’s denote:
\[ t = \tan\left(\frac{v}{2}\right) \]
Then, we can express \(\cos(v)\) and \(\sin(v)\) in terms of \(t\):
\[ \cos(v) = \frac{1 - t^2}{1 + t^2}, \quad \sin(v) = \frac{2t}{1 + t^2} \]
Substituting these into the integral will allow us to integrate more easily.

### Step 7: Back substitute for \(y\)
Once we have the integral solved, we will back substitute \(v = x + y\) to express our solution in terms of \(x\) and \(y\).

### Final Solution
After performing the integration and back substitution, we will arrive at the final solution of the differential equation.

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