What is the solution of the differential equation x dy - y dx = 0 ?

Options

A. xy = c
B. y = cx
C. x + y = c
D. x - y = c

Answer

To solve the differential equation \( x \, dy - y \, dx = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ x \, dy - y \, dx = 0 \]
We can rearrange this to isolate the differentials:
\[ x \, dy = y \, dx \]

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

### Step 3: Integrate 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: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:
\[ |y| = e^{\ln |x| + C} = |x| e^C \]
Let \( k = e^C \), where \( k \) is a positive constant. Thus, we can write:
\[ y = kx \]

### Step 5: General solution
The general solution can be expressed as:
\[ y = Cx \]
where \( C \) is a constant that can take any real value.

### Final Result
Thus, the solution of the differential equation \( x \, dy - y \, dx = 0 \) is:
\[ y = Cx \]

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