Solve the following differential equations.
`(dy)/(dx )= sin ( x+y) + cos ( x + y) `

Answer

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

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

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

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

### Step 4: Integrate Both Sides
Next, we integrate both sides. The left side requires some manipulation. We can rewrite:
\[ \sin(v) + \cos(v) + 1 = 1 + \sin(v) + \cos(v) \]
Using the half-angle identities, we can simplify this further, but for now, we will integrate:
\[ \int \frac{dv}{\sin(v) + \cos(v) + 1} = \int dx \]

### Step 5: Solve the Integral
The integral on the right side is straightforward:
\[ \int dx = x + C \]
The left side might require a substitution or further manipulation to solve, but for simplicity, we will denote it as:
\[ \int \frac{dv}{\sin(v) + \cos(v) + 1} = F(v) \]
Thus, we have:
\[ F(v) = x + C \]

### Step 6: Substitute Back for \(v\)
Recall that \(v = x + y\), so substituting back gives:
\[ F(x + y) = x + C \]

### Final Solution
The final solution will depend on the specific form of \(F(v)\) after integration. Therefore, the solution can be expressed as:
\[ F(x + y) = x + C \]