The solution of the differential equation `xdy=(tan y+(e^(1)//x^(2))/(x)secy)dx` is (where C is the constant of integration)

Options

A. `sin y=e^((1)/(x^(2)))+c`
B. `(2siny)/(x)+e^((1)/(x^(2)))=C`
C. `(siny)/(x)-e^((1)/(x^(2)))=C`
D. `siny-xe^((1)/(x^(2)))=C`

Answer

To solve the differential equation \( x \, dy = \left( \tan y + \frac{e^{1/x^2}}{x} \sec y \right) dx \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rewriting the equation in a more manageable form. We can divide both sides by \( x \) to isolate \( dy \):
\[ dy = \left( \frac{\tan y}{x} + \frac{e^{1/x^2}}{x^2} \sec y \right) dx \]

### Step 2: Simplifying the Equation
Next, we can multiply both sides by \( \cos y \) to eliminate the secant function:
\[ \cos y \, dy = \left( \frac{\sin y}{x} + \frac{e^{1/x^2}}{x^2} \right) dx \]

### Step 3: Separating Variables
Now we can rearrange the equation to separate the variables \( y \) and \( x \):
\[ \cos y \, dy - \frac{\sin y}{x} \, dx = \frac{e^{1/x^2}}{x^2} \, dx \]

### Step 4: Integrating Both Sides
We can now integrate both sides. The left side can be integrated with respect to \( y \) and the right side with respect to \( x \):
\[ \int \cos y \, dy - \int \frac{\sin y}{x} \, dx = \int \frac{e^{1/x^2}}{x^2} \, dx \]

The left side integrates to:
\[ \sin y - \frac{\sin y}{x} = \sin y \left( 1 - \frac{1}{x} \right) \]

The right side requires integration by substitution. Let \( u = \frac{1}{x^2} \), then \( du = -\frac{2}{x^3} \, dx \), which gives us:
\[ -\frac{1}{2} e^{1/x^2} + C \]

### Step 5: Combining Results
Combining the results from the integrations, we have:
\[ \sin y \left( 1 - \frac{1}{x} \right) = -\frac{1}{2} e^{1/x^2} + C \]

### Final Step: Rearranging the Equation
Rearranging gives us the final solution:
\[ 2 \sin y \left( 1 - \frac{1}{x} \right) = -e^{1/x^2} + C \]

### Summary of the Solution
Thus, the solution of the differential equation is:
\[ 2 \sin y \left( 1 - \frac{1}{x} \right) + e^{1/x^2} = C \]