The solution of the differential equation `(dy)/(dx)=(x-y)/(x+4y)` is (where C is the constant of integration)

Options

A. `xy+y^(2)=x+C`
B. `xy-y^(2)=x^(2)+C`
C. `xy+2y^(2)=x^(2)+C`
D. `2xy+4y^(2)=x^(2)+C`

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{x - y}{x + 4y}\), we will follow these steps:

### Step 1: Cross-multiply
We start by cross-multiplying to separate the variables:
\[ dy \cdot (x + 4y) = (x - y) \cdot dx \]
This gives us:
\[ (x + 4y) dy = (x - y) dx \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to group terms involving \(y\) on one side and terms involving \(x\) on the other side:
\[ x dy + 4y dy + y dx = x dx \]
This can be rewritten as:
\[ x dy + 4y dy + y dx - x dx = 0 \]
or
\[ x dy + (4y + y) dy = x dx \]
which simplifies to:
\[ x dy + 5y dy = x dx \]

### Step 3: Integrate both sides
Now we can integrate both sides. We will integrate the left side with respect to \(y\) and the right side with respect to \(x\):
\[ \int (x + 5y) dy = \int x dx \]
This gives us:
\[ xy + \frac{5y^2}{2} = \frac{x^2}{2} + C \]

### Step 4: Rearranging the equation
To express the solution in a more standard form, we can rearrange the equation:
\[ 2xy + 5y^2 = x^2 + 2C \]
Letting \(C' = 2C\) (since \(C\) is an arbitrary constant), we can write:
\[ 2xy + 5y^2 = x^2 + C' \]

### Final Solution
Thus, the solution of the differential equation is:
\[ 2xy + 5y^2 = x^2 + C \]
where \(C\) is the constant of integration.

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