What is the general solution of the differential equation `(dy)/(dx) + (x)/(y) = 0` ?

Options

A. `x^(2) + y^(2)= c`
B. `x^(2) - y^(2) = c`
C. `x^(2) + y^(2) = cxy`
D. `x + y = c`

Answer

To find the general solution of the differential equation

\[ \frac{dy}{dx} + \frac{x}{y} = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = -\frac{x}{y}. \]

**Hint:** Isolate the derivative on one side to facilitate separation of variables.

### Step 2: Separating Variables
Next, we separate the variables \(y\) and \(x\). We can do this by multiplying both sides by \(y\) and \(dx\):

\[ y \, dy = -x \, dx. \]

**Hint:** Multiply both sides by the appropriate terms to group \(y\) terms on one side and \(x\) terms on the other.

### Step 3: Integrating Both Sides
Now we integrate both sides:

\[ \int y \, dy = \int -x \, dx. \]

The left side integrates to:

\[ \frac{y^2}{2} + C_1, \]

and the right side integrates to:

\[ -\frac{x^2}{2} + C_2. \]

**Hint:** Remember to add the constant of integration after integrating both sides.

### Step 4: Combining Constants
We can combine the constants of integration \(C_1\) and \(C_2\) into a single constant \(C\):

\[ \frac{y^2}{2} = -\frac{x^2}{2} + C. \]

**Hint:** Combining constants simplifies the equation and makes it easier to express the solution.

### Step 5: Rearranging the Equation
Multiplying through by 2 to eliminate the fractions gives us:

\[ y^2 = -x^2 + 2C. \]

Rearranging this, we have:

\[ x^2 + y^2 = 2C. \]

**Hint:** Rearranging the equation can help identify the form of the solution.

### Step 6: Final Form of the General Solution
Finally, we can express the equation in a more standard form:

\[ x^2 + y^2 = C, \]

where \(C = 2C\) is still a constant.

**Hint:** The final equation represents a family of circles centered at the origin.

### Conclusion
The general solution of the differential equation \(\frac{dy}{dx} + \frac{x}{y} = 0\) is:

\[ x^2 + y^2 = C, \]

where \(C\) is a constant.