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

Options

A. xy = c
B. x = cy
C. y = cx
D. None of these

Answer

To solve the differential equation \(\frac{dy}{dx} + \frac{y}{x} = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the given equation:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 2: Separating Variables
Next, we separate the variables \(y\) and \(x\):
\[ \frac{dy}{y} = -\frac{dx}{x} \]

### Step 3: Integrating Both Sides
Now, we integrate both sides:
\[ \int \frac{dy}{y} = \int -\frac{dx}{x} \]
This gives us:
\[ \ln |y| = -\ln |x| + C \]
where \(C\) is the constant of integration.

### Step 4: Simplifying the Equation
We can simplify the equation using properties of logarithms:
\[ \ln |y| + \ln |x| = C \]
This can be rewritten as:
\[ \ln |xy| = C \]

### Step 5: Exponentiating Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ |xy| = e^C \]
Let \(k = e^C\), where \(k\) is a positive constant. Thus, we have:
\[ xy = k \]

### Step 6: General Solution
The general solution of the differential equation can be expressed as:
\[ xy = C \]
where \(C\) is a constant.

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

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