`(dy)/(dx)=sin(x+y)+cos(x+y),` where `x+y=u`

Options

A. `1+tanu=e^(x)+c`
B. `1+tan((u)/(2))=e^(x)+c`
C. `1+tan((u)/(2))=ce^(x)`
D. `tan(x+y)=ce^(x)`

Answer

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

### Step 1: Substitute \(u = x + y\)
We start by substituting \(u = x + y\). This gives us:
\[ y = u - x \]
Now, differentiate both sides with respect to \(x\):
\[ \frac{dy}{dx} = \frac{du}{dx} - 1 \]

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

### Step 3: Separate variables
We can separate the variables:
\[ \frac{du}{\sin(u) + \cos(u) + 1} = dx \]

### Step 4: Integrate both sides
Now we integrate both sides. The left side requires integrating with respect to \(u\) and the right side with respect to \(x\):
\[ \int \frac{du}{\sin(u) + \cos(u) + 1} = \int dx \]

### Step 5: Simplify the left side
To simplify the left side, we can use the identity:
\[ \sin(u) + \cos(u) = \sqrt{2} \sin\left(u + \frac{\pi}{4}\right) \]
However, for integration, we can directly integrate:
\[ \int \frac{du}{\sin(u) + \cos(u) + 1} \]
This integral can be solved using trigonometric identities or numerical methods, but for simplicity, we will denote the result as \(F(u)\).

### Step 6: Solve the integral
After integrating, we have:
\[ F(u) = x + C \]
where \(C\) is the constant of integration.

### Step 7: Substitute back for \(u\)
Recall that \(u = x + y\), so we substitute back:
\[ F(x + y) = x + C \]

### Step 8: Final expression
The final expression will depend on the form of \(F(u)\) obtained from the integration.