Solution of differential equation `(dy)/(dx)=sin(x+y)+cos(x+y)` is

Options

A. `log|1+tan.(x+y)/(2)|=y+c`
B. `log|2+sec.(x+y)/(2)|=x+c`
C. `log|1+tan(x+y)|=y+c`
D. none of these

Answer

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

### Step 1: Substitute \( t = x + y \)
Let \( t = x + y \). Then, we can express \( y \) in terms of \( t \):
\[ y = t - x \]

### Step 2: Differentiate \( t \) with respect to \( x \)
Now, differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \frac{d}{dx}(x + y) = 1 + \frac{dy}{dx} \]

### Step 3: Substitute \(\frac{dy}{dx}\) in the original equation
From the original equation, we have:
\[ \frac{dy}{dx} = \sin(t) + \cos(t) \]
Substituting this into the differentiated equation gives:
\[ \frac{dt}{dx} = 1 + \sin(t) + \cos(t) \]

### Step 4: Rearranging the equation
Rearranging the equation, we have:
\[ \frac{dt}{dx} = 1 + \sin(t) + \cos(t) \]

### Step 5: Separate variables
We can separate variables:
\[ \frac{dt}{1 + \sin(t) + \cos(t)} = dx \]

### Step 6: Integrate both sides
Now, we need to integrate both sides. The left side requires some manipulation:
\[ 1 + \sin(t) + \cos(t) = 1 + \sqrt{2}\sin\left(t + \frac{\pi}{4}\right) \]
This can be integrated using standard techniques.

### Step 7: Solve the integral
The integral on the left-hand side can be computed, and we will denote the result as \( F(t) \):
\[ \int \frac{dt}{1 + \sin(t) + \cos(t)} = \int dx \]

### Step 8: Solve for \( y \)
After integrating, we will have a function of \( t \) on the left and \( x \) on the right. We can then solve for \( y \) by substituting back \( t = x + y \).

### Final Solution
The final solution will be in the form of \( y \) expressed in terms of \( x \) and a constant of integration.

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