The solution of the differential equation
`(x+y)^(2)(dy)/(dx) = a^(2) ` is

Options

A. `(x+y)^(2) = (a^(2)x)/2 + C`
B. ` (x+y)^(2) = a^(2)x +C`
C. `(x+y)^(2) = 2a^(2)x +C`
D. None of these

Explanation

To solve the differential equation \((x+y)^{2} \frac{dy}{dx} = a^{2}\), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given differential equation:
\[ (x+y)^{2} \frac{dy}{dx} = a^{2} \]
We can rewrite this as:
\[ \frac{dy}{dx} = \frac{a^{2}}{(x+y)^{2}} \]

### Step 2: Substitute Variables
Let \(v = x + y\). Then, we can express \(y\) in terms of \(v\):
\[ y = v - x \]
Now differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{dv}{dx} - 1 \]

### Step 3: Substitute \(\frac{dy}{dx}\) in the Equation
Substituting \(\frac{dy}{dx}\) back into the equation gives:
\[ \frac{dv}{dx} - 1 = \frac{a^{2}}{v^{2}} \]
Rearranging this, we have:
\[ \frac{dv}{dx} = \frac{a^{2}}{v^{2}} + 1 \]

### Step 4: Separate Variables
Now we separate the variables:
\[ \frac{dv}{\frac{a^{2}}{v^{2}} + 1} = dx \]

### Step 5: Integrate Both Sides
Next, we integrate both sides. The left side requires some manipulation:
\[ \frac{dv}{\frac{a^{2}}{v^{2}} + 1} = \frac{v^{2}}{a^{2} + v^{2}} dv \]
Now, we integrate:
\[ \int \frac{v^{2}}{a^{2} + v^{2}} dv = \int dx \]

### Step 6: Solve the Integrals
The integral on the left can be solved using the formula:
\[ \int \frac{v^{2}}{a^{2} + v^{2}} dv = v - a \tan^{-1}\left(\frac{v}{a}\right) + C \]
The integral on the right is simply:
\[ x + C \]

### Step 7: Combine the Results
Equating both sides gives:
\[ v - a \tan^{-1}\left(\frac{v}{a}\right) = x + C \]
Substituting back \(v = x + y\):
\[ (x + y) - a \tan^{-1}\left(\frac{x + y}{a}\right) = x + C \]

### Step 8: Simplify the Equation
Rearranging the equation gives:
\[ y - a \tan^{-1}\left(\frac{x + y}{a}\right) = C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = a \tan^{-1}\left(\frac{x + y}{a}\right) + C \]