A curve passes through `(1,pi/4)` and at `(x,y)` its slope is `(sin 2y)/(x+tan y).` Find the equation to the curve.

Options

A. ` x = tan y`
B. `y= tan x`
C. ` x=2 tan Y`
D. `y=2 tan x`

Answer

To find the equation of the curve that passes through the point \( (1, \frac{\pi}{4}) \) and has a slope given by \( \frac{\sin 2y}{x + \tan y} \), we will follow these steps:

### Step 1: Set up the differential equation
The slope of the curve is given by:
\[ \frac{dy}{dx} = \frac{\sin 2y}{x + \tan y} \]

### Step 2: Rearranging the equation
We can rearrange this to express \( dx \) in terms of \( dy \):
\[ dx = \frac{x + \tan y}{\sin 2y} dy \]

### Step 3: Separate variables
We can separate the variables \( x \) and \( y \):
\[ \frac{dx}{x + \tan y} = \frac{dy}{\sin 2y} \]

### Step 4: Integrate both sides
Now we will integrate both sides. The left side requires the integration of \( \frac{1}{x + \tan y} \) with respect to \( x \), and the right side requires the integration of \( \frac{1}{\sin 2y} \) with respect to \( y \).

1. **Left side**:
\[ \int \frac{dx}{x + \tan y} = \ln |x + \tan y| + C_1 \]

2. **Right side**:
To integrate the right side, we use the identity \( \sin 2y = 2 \sin y \cos y \):
\[ \int \frac{dy}{\sin 2y} = \int \frac{1}{2 \sin y \cos y} dy = \frac{1}{2} \int \frac{1}{\sin y \cos y} dy \]
This can be solved using the substitution \( u = \tan y \):
\[ \int \frac{1}{\sin y \cos y} dy = \ln |\tan y + \sec y| + C_2 \]

### Step 5: Combine the results
Combining the results from both sides gives us:
\[ \ln |x + \tan y| = \frac{1}{2} \ln |\tan y + \sec y| + C \]

### Step 6: Exponentiate both sides
Exponentiating both sides results in:
\[ |x + \tan y| = K |\tan y + \sec y|^{1/2} \]
where \( K = e^C \).

### Step 7: Use the initial condition
Now we use the initial condition \( (1, \frac{\pi}{4}) \):
\[ x = 1, \quad y = \frac{\pi}{4} \Rightarrow \tan\left(\frac{\pi}{4}\right) = 1, \quad \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \]
Substituting these values:
\[ |1 + 1| = K |\tan\left(\frac{\pi}{4}\right) + \sec\left(\frac{\pi}{4}\right)|^{1/2} \]
\[ 2 = K |1 + \sqrt{2}|^{1/2} \]

### Step 8: Solve for \( K \)
Solving for \( K \):
\[ K = \frac{2}{\sqrt{1 + \sqrt{2}}} \]

### Final Equation
Substituting \( K \) back into the equation gives us the final equation of the curve.