General solution of `(dy)/(dx)+(x)/(y)=0` is

Options

A. `x^(2)+y^(2)+c=0`
B. `x^(2)-y^(2)+c=0`
C. `x^(2)+y+c=0`
D. `x^(2)-y+c=0`

Answer

To find the general solution of the differential equation \(\frac{dy}{dx} + \frac{x}{y} = 0\), we can follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ \frac{dy}{dx} + \frac{x}{y} = 0 \]
We can isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{x}{y} \]

**Hint:** Rearranging the equation helps to express the derivative in terms of \(x\) and \(y\).

### Step 2: Separate the variables
Next, we can separate the variables \(y\) and \(x\):
\[ y \, dy = -x \, dx \]

**Hint:** Separating variables allows us to integrate both sides independently.

### Step 3: Integrate both sides
Now we integrate both sides:
\[ \int y \, dy = \int -x \, dx \]
The left side integrates to:
\[ \frac{y^2}{2} + C_1 \]
The right side integrates to:
\[ -\frac{x^2}{2} + C_2 \]
Combining these, we get:
\[ \frac{y^2}{2} = -\frac{x^2}{2} + C \]
where \(C = C_2 - C_1\).

**Hint:** Remember to include the constant of integration when integrating.

### Step 4: Rearranging the equation
Multiply through by 2 to eliminate the fractions:
\[ y^2 = -x^2 + 2C \]
Rearranging gives:
\[ x^2 + y^2 = 2C \]

**Hint:** Rearranging the equation helps in identifying the standard form of the solution.

### Step 5: General solution
Let \(C' = 2C\) (where \(C'\) is just another constant), we can write the general solution as:
\[ x^2 + y^2 + C' = 0 \]
This can also be expressed as:
\[ x^2 + y^2 + C = 0 \]
where \(C\) is a constant.

**Hint:** The general solution often takes the form of an equation involving constants.

### Final Answer
The general solution of the differential equation \(\frac{dy}{dx} + \frac{x}{y} = 0\) is:
\[ x^2 + y^2 + C = 0 \]