A curve passes through the point `(1,pi/6)` . Let the slope of the curve at each point `(x , y)` be `y/x+sec(y/x),x > 0.` Then the equation of the curve is

Options

A. `sin((y)/(x))=logx+(1)/(2)`
B. `"cosec"((y)/(x))=logx+2`
C. `sec((2y)/(x))=logx+2`
D. `cos((2y)/(x))=logx+(1)/(2)`

Explanation

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step 1: Set up the differential equation
We are given that the slope of the curve at each point \((x, y)\) is given by:
\[ \frac{dy}{dx} = \frac{y}{x} + \sec\left(\frac{y}{x}\right) \]
This is our starting point.

### Step 2: Substitute \(y = vx\)
Let us make the substitution \(y = vx\), where \(v\) is a function of \(x\). Then, we can express \(\frac{y}{x}\) as \(v\).

### Step 3: Differentiate \(y\) with respect to \(x\)
Using the product rule, we differentiate \(y = vx\):
\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]

### Step 4: Substitute into the differential equation
Substituting \(y = vx\) into the differential equation gives:
\[ v + x\frac{dv}{dx} = v + \sec(v) \]
We can simplify this to:
\[ x\frac{dv}{dx} = \sec(v) \]

### Step 5: Separate the variables
Now, we separate the variables:
\[ \frac{dv}{\sec(v)} = \frac{dx}{x} \]
Using the identity \(\sec(v) = \frac{1}{\cos(v)}\), we rewrite the left side:
\[ \cos(v) dv = \frac{dx}{x} \]

### Step 6: Integrate both sides
Now we integrate both sides:
\[ \int \cos(v) dv = \int \frac{dx}{x} \]
The left side integrates to \(\sin(v)\), and the right side integrates to \(\ln|x| + C\):
\[ \sin(v) = \ln|x| + C \]

### Step 7: Substitute back for \(v\)
Recall that \(v = \frac{y}{x}\), so we substitute back:
\[ \sin\left(\frac{y}{x}\right) = \ln|x| + C \]

### Step 8: Use the initial condition
We know the curve passes through the point \((1, \frac{\pi}{6})\). We substitute \(x = 1\) and \(y = \frac{\pi}{6}\):
\[ \sin\left(\frac{\pi}{6}\right) = \ln(1) + C \]
Since \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\) and \(\ln(1) = 0\), we have:
\[ \frac{1}{2} = 0 + C \implies C = \frac{1}{2} \]

### Step 9: Write the final equation
Substituting \(C\) back into the equation gives:
\[ \sin\left(\frac{y}{x}\right) = \ln|x| + \frac{1}{2} \]

### Final Answer
Thus, the equation of the curve is:
\[ \sin\left(\frac{y}{x}\right) = \ln(x) + \frac{1}{2} \]