`(x(dy)/(dx)-y)e^(y//x)=x^(2)sec^(2)x`

Options

A. `e^(y//x)=tanx+c`
B. `e^(y//x)tanx=c`
C. `e^(y//x)+tanx=c`
D. `e^(-y//x)+tanx=c`

Answer

To solve the differential equation \( (x \frac{dy}{dx} - y)e^{\frac{y}{x}} = x^2 \sec^2 x \), we will follow these steps:

### Step 1: Rewrite the equation
We start by dividing the entire equation by \( x \):
\[ \frac{dy}{dx} - \frac{y}{x} = \frac{x^2 \sec^2 x}{x} e^{-\frac{y}{x}} \]
This simplifies to:
\[ \frac{dy}{dx} - \frac{y}{x} = x \sec^2 x e^{-\frac{y}{x}} \]

### Step 2: Substitute \( v = \frac{y}{x} \)
Let \( v = \frac{y}{x} \). Then, we can express \( y \) as:
\[ y = vx \]
Now, differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

### Step 3: Substitute \( y \) and \( \frac{dy}{dx} \) into the equation
Substituting \( y \) and \( \frac{dy}{dx} \) into the equation gives:
\[ v + x \frac{dv}{dx} - v = x \sec^2 x e^{-v} \]
This simplifies to:
\[ x \frac{dv}{dx} = x \sec^2 x e^{-v} \]

### Step 4: Divide both sides by \( x \)
Assuming \( x \neq 0 \), we can divide both sides by \( x \):
\[ \frac{dv}{dx} = \sec^2 x e^{-v} \]

### Step 5: Separate the variables
Now, we can separate the variables:
\[ e^v dv = \sec^2 x dx \]

### Step 6: Integrate both sides
Integrate both sides:
\[ \int e^v dv = \int \sec^2 x dx \]
The left side integrates to:
\[ e^v = \tan x + C \]

### Step 7: Substitute back for \( v \)
Recall that \( v = \frac{y}{x} \), so we substitute back:
\[ e^{\frac{y}{x}} = \tan x + C \]

### Final Solution
Thus, the solution to the differential equation is:
\[ e^{\frac{y}{x}} = \tan x + C \]