The solution of differential equation `dy/dx=cos(x+y)` is :

Options

A. `tan(x+y)=x+c`
B. `tan((x+y)/(2))=x+c`
C. `tan(x+y)=y+c`
D. None of the above

Answer

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

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

### Step 2: Differentiate
Now, differentiate both sides with respect to \(x\):
\[ \frac{dy}{dx} = \frac{dt}{dx} - 1 \]

### Step 3: Substitute in the Original Equation
Substituting \(\frac{dy}{dx}\) into the original differential equation gives:
\[ \frac{dt}{dx} - 1 = \cos(t) \]
Rearranging this, we find:
\[ \frac{dt}{dx} = \cos(t) + 1 \]

### Step 4: Separate Variables
Now, we can separate the variables:
\[ \frac{dt}{\cos(t) + 1} = dx \]

### Step 5: Integrate Both Sides
Next, we integrate both sides. The left side can be simplified using the identity \(\cos(t) + 1 = 2\cos^2(t/2)\):
\[ \int \frac{dt}{\cos(t) + 1} = \int dx \]
This becomes:
\[ \int \frac{dt}{2\cos^2(t/2)} = \int dx \]
\[ \frac{1}{2} \int \sec^2(t/2) dt = x + C \]

### Step 6: Integrate the Left Side
The integral of \(\sec^2(t/2)\) is:
\[ \frac{1}{2} \tan(t/2) = x + C \]

### Step 7: Substitute Back for \(t\)
Recall that \(t = x + y\), so we have:
\[ \frac{1}{2} \tan\left(\frac{x + y}{2}\right) = x + C \]

### Step 8: Final Rearrangement
Multiplying through by 2 gives:
\[ \tan\left(\frac{x + y}{2}\right) = 2(x + C) \]

This is the implicit solution to the differential equation.