General solution of `(dy)/(dx)=cotx` is

Options

A. `e^(-y)=c cos x`
B. `e^(y)=c cos x`
C. `e^(y)=c sin x`
D. `e^(-y) = c sin x`

Answer

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

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \cot x \]

### Step 2: Separate the variables
We can separate the variables by multiplying both sides by \(dx\):
\[ dy = \cot x \, dx \]

### Step 3: Integrate both sides
Next, we integrate both sides. The left side integrates to \(y\), and we need to find the integral of \(\cot x\):
\[ \int dy = \int \cot x \, dx \]
The integral of \(\cot x\) is \(\ln |\sin x| + C\), where \(C\) is the constant of integration. Thus, we have:
\[ y = \ln |\sin x| + C \]

### Step 4: Simplify the solution
We can express the solution in a more compact form. By exponentiating both sides, we can eliminate the logarithm:
\[ e^y = e^{\ln |\sin x| + C} \]
This simplifies to:
\[ e^y = |\sin x| \cdot e^C \]
Let \(k = e^C\) (where \(k\) is a positive constant), we can write:
\[ e^y = k |\sin x| \]

### Final General Solution
Thus, the general solution of the differential equation is:
\[ y = \ln(k |\sin x|) \]
where \(k\) is a constant.